164.13/58.62 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^2)).

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 CdtNarrowingProof (BOTH BOUNDS(ID, ID))

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↳6 CdtProblem

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

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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 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))

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↳14 CdtProblem

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

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↳16 CdtProblem

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

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↳18 CdtProblem

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

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↳20 CdtProblem

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

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↳22 CdtProblem

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

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↳24 CdtProblem

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

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↳26 CdtProblem

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

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↳28 CdtProblem

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

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↳30 CdtProblem

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

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↳32 CdtProblem

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

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↳34 CdtProblem

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↳35 CdtUnreachableProof (⇔)

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↳36 CdtProblem

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

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↳38 CdtProblem

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

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↳40 CdtProblem

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

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↳42 CdtProblem

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

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↳44 CdtProblem

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

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↳46 CdtProblem

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

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↳48 CdtProblem

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

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↳50 CdtProblem

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

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↳52 BOUNDS(O(1), O(1))

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(0) Obligation:

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

active(from(X)) → mark(cons(X, from(s(X)))) 164.13/58.62
active(sel(0, cons(X, Y))) → mark(X) 164.13/58.62
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z)) 164.13/58.62
active(from(X)) → from(active(X)) 164.13/58.62
active(cons(X1, X2)) → cons(active(X1), X2) 164.13/58.62
active(s(X)) → s(active(X)) 164.13/58.62
active(sel(X1, X2)) → sel(active(X1), X2) 164.13/58.62
active(sel(X1, X2)) → sel(X1, active(X2)) 164.13/58.62
from(mark(X)) → mark(from(X)) 164.13/58.62
cons(mark(X1), X2) → mark(cons(X1, X2)) 164.13/58.62
s(mark(X)) → mark(s(X)) 164.13/58.62
sel(mark(X1), X2) → mark(sel(X1, X2)) 164.13/58.62
sel(X1, mark(X2)) → mark(sel(X1, X2)) 164.13/58.62
proper(from(X)) → from(proper(X)) 164.13/58.62
proper(cons(X1, X2)) → cons(proper(X1), proper(X2)) 164.13/58.62
proper(s(X)) → s(proper(X)) 164.13/58.62
proper(sel(X1, X2)) → sel(proper(X1), proper(X2)) 164.13/58.62
proper(0) → ok(0) 164.13/58.62
from(ok(X)) → ok(from(X)) 164.13/58.62
cons(ok(X1), ok(X2)) → ok(cons(X1, X2)) 164.13/58.62
s(ok(X)) → ok(s(X)) 164.13/58.62
sel(ok(X1), ok(X2)) → ok(sel(X1, X2)) 164.13/58.62
top(mark(X)) → top(proper(X)) 164.13/58.62
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(from(z0)) → mark(cons(z0, from(s(z0)))) 164.13/58.62
active(sel(0, cons(z0, z1))) → mark(z0) 164.13/58.62
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 164.13/58.62
active(from(z0)) → from(active(z0)) 164.13/58.62
active(cons(z0, z1)) → cons(active(z0), z1) 164.13/58.62
active(s(z0)) → s(active(z0)) 164.13/58.62
active(sel(z0, z1)) → sel(active(z0), z1) 164.13/58.62
active(sel(z0, z1)) → sel(z0, active(z1)) 164.13/58.62
from(mark(z0)) → mark(from(z0)) 164.13/58.62
from(ok(z0)) → ok(from(z0)) 164.13/58.62
cons(mark(z0), z1) → mark(cons(z0, z1)) 164.13/58.62
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 164.13/58.62
s(mark(z0)) → mark(s(z0)) 164.13/58.62
s(ok(z0)) → ok(s(z0)) 164.13/58.62
sel(mark(z0), z1) → mark(sel(z0, z1)) 164.13/58.62
sel(z0, mark(z1)) → mark(sel(z0, z1)) 164.13/58.62
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 164.13/58.62
proper(from(z0)) → from(proper(z0)) 164.13/58.62
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 164.13/58.62
proper(s(z0)) → s(proper(z0)) 164.13/58.62
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 164.13/58.62
proper(0) → ok(0) 164.13/58.62
top(mark(z0)) → top(proper(z0)) 164.13/58.62
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(from(z0)) → c(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0)) 164.13/58.62
ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2)) 164.13/58.62
ACTIVE(from(z0)) → c3(FROM(active(z0)), ACTIVE(z0)) 164.13/58.62
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0)) 164.13/58.62
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0)) 164.13/58.62
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0)) 164.13/58.62
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1)) 164.13/58.62
FROM(mark(z0)) → c8(FROM(z0)) 164.13/58.62
FROM(ok(z0)) → c9(FROM(z0)) 164.13/58.62
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.13/58.62
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.13/58.62
S(mark(z0)) → c12(S(z0)) 164.13/58.62
S(ok(z0)) → c13(S(z0)) 164.13/58.62
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.13/58.62
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.13/58.62
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1)) 164.13/58.62
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0)) 164.13/58.65
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 164.13/58.65
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0)) 164.13/58.65
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 164.13/58.65
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 164.13/58.65
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(from(z0)) → c(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0)) 164.13/58.65
ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2)) 164.13/58.65
ACTIVE(from(z0)) → c3(FROM(active(z0)), ACTIVE(z0)) 164.13/58.65
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0)) 164.13/58.65
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0)) 164.13/58.65
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0)) 164.13/58.65
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1)) 164.13/58.65
FROM(mark(z0)) → c8(FROM(z0)) 164.13/58.65
FROM(ok(z0)) → c9(FROM(z0)) 164.13/58.65
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.13/58.65
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.13/58.65
S(mark(z0)) → c12(S(z0)) 164.13/58.65
S(ok(z0)) → c13(S(z0)) 164.13/58.65
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.13/58.65
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.13/58.65
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1)) 164.13/58.65
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0)) 164.13/58.65
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 164.13/58.65
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0)) 164.13/58.65
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 164.13/58.65
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 164.13/58.65
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, PROPER, TOP

Compound Symbols:

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

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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(from(z0)) → mark(cons(z0, from(s(z0)))) 164.13/58.65
active(sel(0, cons(z0, z1))) → mark(z0) 164.13/58.65
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 164.13/58.65
active(from(z0)) → from(active(z0)) 164.13/58.65
active(cons(z0, z1)) → cons(active(z0), z1) 164.13/58.65
active(s(z0)) → s(active(z0)) 164.13/58.65
active(sel(z0, z1)) → sel(active(z0), z1) 164.13/58.65
active(sel(z0, z1)) → sel(z0, active(z1)) 164.13/58.65
from(mark(z0)) → mark(from(z0)) 164.13/58.65
from(ok(z0)) → ok(from(z0)) 164.13/58.65
cons(mark(z0), z1) → mark(cons(z0, z1)) 164.13/58.65
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 164.13/58.65
s(mark(z0)) → mark(s(z0)) 164.13/58.65
s(ok(z0)) → ok(s(z0)) 164.13/58.65
sel(mark(z0), z1) → mark(sel(z0, z1)) 164.13/58.65
sel(z0, mark(z1)) → mark(sel(z0, z1)) 164.13/58.65
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 164.13/58.65
proper(from(z0)) → from(proper(z0)) 164.13/58.65
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 164.13/58.65
proper(s(z0)) → s(proper(z0)) 164.13/58.65
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 164.13/58.65
proper(0) → ok(0) 164.13/58.65
top(mark(z0)) → top(proper(z0)) 164.13/58.65
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2)) 164.13/58.65
ACTIVE(from(z0)) → c3(FROM(active(z0)), ACTIVE(z0)) 164.13/58.65
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0)) 164.13/58.65
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0)) 164.13/58.65
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0)) 164.13/58.65
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1)) 164.13/58.65
FROM(mark(z0)) → c8(FROM(z0)) 164.13/58.65
FROM(ok(z0)) → c9(FROM(z0)) 164.13/58.65
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.13/58.65
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.13/58.65
S(mark(z0)) → c12(S(z0)) 164.13/58.65
S(ok(z0)) → c13(S(z0)) 164.13/58.65
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.13/58.65
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.13/58.65
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1)) 164.13/58.65
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0)) 164.13/58.65
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 164.13/58.65
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0)) 164.13/58.65
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 164.13/58.65
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 164.13/58.65
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 164.13/58.65
ACTIVE(from(z0)) → c(S(z0))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2)) 164.50/58.75
ACTIVE(from(z0)) → c3(FROM(active(z0)), ACTIVE(z0)) 164.50/58.75
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0)) 164.50/58.75
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0)) 164.50/58.75
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0)) 164.50/58.75
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1)) 164.50/58.75
FROM(mark(z0)) → c8(FROM(z0)) 164.50/58.75
FROM(ok(z0)) → c9(FROM(z0)) 164.50/58.75
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.50/58.75
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.50/58.75
S(mark(z0)) → c12(S(z0)) 164.50/58.75
S(ok(z0)) → c13(S(z0)) 164.50/58.75
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.50/58.75
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.50/58.75
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1)) 164.50/58.75
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0)) 164.50/58.75
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 164.50/58.75
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0)) 164.50/58.75
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 164.50/58.75
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 164.50/58.75
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 164.50/58.75
ACTIVE(from(z0)) → c(S(z0))

K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, PROPER, TOP

Compound Symbols:

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

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

Use narrowing to replace ACTIVE(from(z0)) → c3(FROM(active(z0)), ACTIVE(z0)) by

ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.50/58.75
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.50/58.75
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.50/58.75
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 164.50/58.75
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.50/58.75
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 164.50/58.75
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.50/58.75
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))

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(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 164.50/58.75
active(sel(0, cons(z0, z1))) → mark(z0) 164.50/58.75
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 164.50/58.75
active(from(z0)) → from(active(z0)) 164.50/58.75
active(cons(z0, z1)) → cons(active(z0), z1) 164.50/58.75
active(s(z0)) → s(active(z0)) 164.50/58.75
active(sel(z0, z1)) → sel(active(z0), z1) 164.50/58.75
active(sel(z0, z1)) → sel(z0, active(z1)) 164.50/58.75
from(mark(z0)) → mark(from(z0)) 164.50/58.75
from(ok(z0)) → ok(from(z0)) 164.50/58.75
cons(mark(z0), z1) → mark(cons(z0, z1)) 164.50/58.75
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 164.50/58.75
s(mark(z0)) → mark(s(z0)) 164.50/58.75
s(ok(z0)) → ok(s(z0)) 164.50/58.75
sel(mark(z0), z1) → mark(sel(z0, z1)) 164.50/58.75
sel(z0, mark(z1)) → mark(sel(z0, z1)) 164.50/58.75
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 164.50/58.75
proper(from(z0)) → from(proper(z0)) 164.50/58.75
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 164.50/58.75
proper(s(z0)) → s(proper(z0)) 164.50/58.75
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 164.50/58.75
proper(0) → ok(0) 164.50/58.75
top(mark(z0)) → top(proper(z0)) 164.50/58.75
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2)) 164.50/58.75
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0)) 164.50/58.75
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0)) 164.50/58.75
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0)) 164.50/58.75
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1)) 164.50/58.75
FROM(mark(z0)) → c8(FROM(z0)) 164.50/58.75
FROM(ok(z0)) → c9(FROM(z0)) 164.50/58.75
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.50/58.75
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.50/58.75
S(mark(z0)) → c12(S(z0)) 164.50/58.75
S(ok(z0)) → c13(S(z0)) 164.50/58.75
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.50/58.75
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.50/58.75
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1)) 164.50/58.75
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0)) 164.50/58.75
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 164.50/58.75
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0)) 164.50/58.75
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 164.50/58.75
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 164.50/58.75
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 164.50/58.75
ACTIVE(from(z0)) → c(S(z0)) 164.50/58.75
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.50/58.75
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.50/58.75
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.50/58.75
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 164.50/58.75
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.50/58.75
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 164.50/58.75
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.50/58.75
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2)) 164.50/58.75
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0)) 164.50/58.75
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0)) 164.50/58.75
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0)) 164.50/58.75
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1)) 164.50/58.75
FROM(mark(z0)) → c8(FROM(z0)) 164.50/58.75
FROM(ok(z0)) → c9(FROM(z0)) 164.50/58.75
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.50/58.75
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.50/58.75
S(mark(z0)) → c12(S(z0)) 164.50/58.75
S(ok(z0)) → c13(S(z0)) 164.50/58.75
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.50/58.75
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.50/58.75
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1)) 164.50/58.75
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0)) 164.50/58.75
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 164.50/58.75
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0)) 164.50/58.75
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 164.50/58.75
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 164.50/58.75
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 164.50/58.75
ACTIVE(from(z0)) → c(S(z0)) 164.50/58.75
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.50/58.75
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.50/58.75
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.50/58.75
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 164.50/58.75
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.50/58.75
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 164.50/58.75
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.50/58.75
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))

K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, PROPER, TOP

Compound Symbols:

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

164.50/58.75

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

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

ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 164.50/58.75
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 164.50/58.75
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.50/58.75
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 164.50/58.75
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 164.50/58.75
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 164.50/58.75
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 164.50/58.75
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))

164.50/58.75

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 164.50/58.75
active(sel(0, cons(z0, z1))) → mark(z0) 164.50/58.75
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 164.50/58.75
active(from(z0)) → from(active(z0)) 164.50/58.75
active(cons(z0, z1)) → cons(active(z0), z1) 164.50/58.75
active(s(z0)) → s(active(z0)) 164.50/58.75
active(sel(z0, z1)) → sel(active(z0), z1) 164.50/58.75
active(sel(z0, z1)) → sel(z0, active(z1)) 164.50/58.75
from(mark(z0)) → mark(from(z0)) 164.50/58.75
from(ok(z0)) → ok(from(z0)) 164.50/58.75
cons(mark(z0), z1) → mark(cons(z0, z1)) 164.50/58.75
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 164.50/58.75
s(mark(z0)) → mark(s(z0)) 164.50/58.75
s(ok(z0)) → ok(s(z0)) 164.50/58.75
sel(mark(z0), z1) → mark(sel(z0, z1)) 164.50/58.75
sel(z0, mark(z1)) → mark(sel(z0, z1)) 164.50/58.75
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 164.50/58.75
proper(from(z0)) → from(proper(z0)) 164.50/58.75
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 164.50/58.75
proper(s(z0)) → s(proper(z0)) 164.50/58.75
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 164.50/58.75
proper(0) → ok(0) 164.50/58.75
top(mark(z0)) → top(proper(z0)) 164.50/58.75
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2)) 164.50/58.75
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0)) 164.50/58.75
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0)) 164.50/58.75
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1)) 164.50/58.75
FROM(mark(z0)) → c8(FROM(z0)) 164.50/58.75
FROM(ok(z0)) → c9(FROM(z0)) 164.50/58.75
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.50/58.75
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.50/58.75
S(mark(z0)) → c12(S(z0)) 164.50/58.75
S(ok(z0)) → c13(S(z0)) 164.50/58.75
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.50/58.75
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.50/58.75
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1)) 164.50/58.75
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0)) 164.50/58.75
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 164.50/58.75
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0)) 164.50/58.75
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 164.50/58.75
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 164.50/58.75
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 164.50/58.75
ACTIVE(from(z0)) → c(S(z0)) 164.50/58.75
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.50/58.75
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.50/58.75
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.50/58.75
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 164.50/58.75
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.50/58.75
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 164.50/58.75
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.50/58.75
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.50/58.75
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 164.50/58.75
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 164.50/58.75
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.50/58.75
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 164.50/58.75
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 164.50/58.75
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 164.50/58.75
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 164.50/58.75
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2)) 164.50/58.75
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0)) 164.50/58.75
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0)) 164.50/58.75
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1)) 164.50/58.75
FROM(mark(z0)) → c8(FROM(z0)) 164.50/58.75
FROM(ok(z0)) → c9(FROM(z0)) 164.50/58.75
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.50/58.75
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.50/58.75
S(mark(z0)) → c12(S(z0)) 164.50/58.75
S(ok(z0)) → c13(S(z0)) 164.50/58.75
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.50/58.75
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.50/58.75
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1)) 164.50/58.75
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0)) 164.50/58.75
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 164.50/58.75
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0)) 164.50/58.75
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 164.50/58.75
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 164.50/58.75
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 164.50/58.75
ACTIVE(from(z0)) → c(S(z0)) 164.50/58.75
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.50/58.75
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.50/58.75
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.50/58.75
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 164.50/58.75
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.50/58.75
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 164.50/58.75
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.50/58.75
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.50/58.75
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 164.50/58.75
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 164.50/58.75
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.50/58.75
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 164.50/58.75
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 164.50/58.75
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 164.50/58.75
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 164.50/58.75
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))

K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, PROPER, TOP

Compound Symbols:

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

164.50/58.75

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

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

ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.50/58.75
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.50/58.75
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.50/58.75
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 164.50/58.75
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.50/58.75
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 164.50/58.75
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.50/58.75
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))

164.50/58.75

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 164.50/58.75
active(sel(0, cons(z0, z1))) → mark(z0) 164.50/58.75
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 164.50/58.75
active(from(z0)) → from(active(z0)) 164.50/58.75
active(cons(z0, z1)) → cons(active(z0), z1) 164.50/58.75
active(s(z0)) → s(active(z0)) 164.50/58.75
active(sel(z0, z1)) → sel(active(z0), z1) 164.50/58.75
active(sel(z0, z1)) → sel(z0, active(z1)) 164.50/58.75
from(mark(z0)) → mark(from(z0)) 164.50/58.75
from(ok(z0)) → ok(from(z0)) 164.50/58.75
cons(mark(z0), z1) → mark(cons(z0, z1)) 164.50/58.75
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 164.50/58.75
s(mark(z0)) → mark(s(z0)) 164.50/58.75
s(ok(z0)) → ok(s(z0)) 164.50/58.75
sel(mark(z0), z1) → mark(sel(z0, z1)) 164.50/58.75
sel(z0, mark(z1)) → mark(sel(z0, z1)) 164.50/58.75
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 164.50/58.75
proper(from(z0)) → from(proper(z0)) 164.50/58.75
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 164.50/58.75
proper(s(z0)) → s(proper(z0)) 164.50/58.75
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 164.50/58.75
proper(0) → ok(0) 164.50/58.75
top(mark(z0)) → top(proper(z0)) 164.50/58.75
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2)) 164.50/58.75
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0)) 164.50/58.75
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1)) 164.50/58.75
FROM(mark(z0)) → c8(FROM(z0)) 164.50/58.75
FROM(ok(z0)) → c9(FROM(z0)) 164.50/58.75
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.50/58.75
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.50/58.75
S(mark(z0)) → c12(S(z0)) 164.50/58.75
S(ok(z0)) → c13(S(z0)) 164.50/58.75
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.50/58.75
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.50/58.75
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1)) 164.50/58.75
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0)) 164.50/58.75
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 164.50/58.76
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0)) 164.50/58.76
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 164.50/58.76
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 164.50/58.76
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 164.50/58.76
ACTIVE(from(z0)) → c(S(z0)) 164.50/58.76
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.50/58.76
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.50/58.76
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.50/58.76
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 164.50/58.76
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.50/58.76
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 164.50/58.76
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.50/58.76
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.50/58.76
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 164.50/58.76
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 164.50/58.76
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.50/58.76
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 164.50/58.76
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 164.50/58.76
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 164.50/58.76
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 164.50/58.76
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 164.50/58.76
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.50/58.76
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.50/58.76
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.50/58.76
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 164.50/58.76
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.50/58.76
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 164.50/58.76
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.50/58.76
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2)) 164.50/58.76
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0)) 164.50/58.76
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1)) 164.50/58.76
FROM(mark(z0)) → c8(FROM(z0)) 164.50/58.76
FROM(ok(z0)) → c9(FROM(z0)) 164.50/58.76
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.50/58.76
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.50/58.76
S(mark(z0)) → c12(S(z0)) 164.50/58.76
S(ok(z0)) → c13(S(z0)) 164.50/58.76
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.50/58.76
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.50/58.76
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1)) 164.50/58.76
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0)) 164.50/58.76
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 164.50/58.76
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0)) 164.50/58.76
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 164.50/58.76
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 164.50/58.76
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 164.50/58.76
ACTIVE(from(z0)) → c(S(z0)) 164.50/58.76
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.50/58.76
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.50/58.76
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.50/58.76
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 164.50/58.76
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.50/58.76
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 164.50/58.76
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.50/58.76
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.50/58.76
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 164.50/58.76
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 164.50/58.76
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.50/58.76
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 164.50/58.76
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 164.50/58.76
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 164.50/58.76
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 164.50/58.78
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 164.50/58.78
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.50/58.78
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.50/58.78
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.50/58.78
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 164.50/58.78
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.50/58.78
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 164.50/58.78
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.50/58.78
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))

K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, PROPER, TOP

Compound Symbols:

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

164.50/58.78

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

Use narrowing to replace PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0)) by

PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 164.50/58.78
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.50/58.78
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 164.50/58.78
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.50/58.78
PROPER(from(0)) → c17(FROM(ok(0)), PROPER(0))

164.50/58.78

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 164.50/58.78
active(sel(0, cons(z0, z1))) → mark(z0) 164.50/58.78
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 164.50/58.78
active(from(z0)) → from(active(z0)) 164.50/58.78
active(cons(z0, z1)) → cons(active(z0), z1) 164.50/58.78
active(s(z0)) → s(active(z0)) 164.50/58.78
active(sel(z0, z1)) → sel(active(z0), z1) 164.50/58.78
active(sel(z0, z1)) → sel(z0, active(z1)) 164.50/58.78
from(mark(z0)) → mark(from(z0)) 164.50/58.78
from(ok(z0)) → ok(from(z0)) 164.50/58.78
cons(mark(z0), z1) → mark(cons(z0, z1)) 164.50/58.78
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 164.50/58.79
s(mark(z0)) → mark(s(z0)) 164.50/58.79
s(ok(z0)) → ok(s(z0)) 164.50/58.79
sel(mark(z0), z1) → mark(sel(z0, z1)) 164.50/58.79
sel(z0, mark(z1)) → mark(sel(z0, z1)) 164.50/58.79
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 164.50/58.79
proper(from(z0)) → from(proper(z0)) 164.50/58.79
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 164.50/58.79
proper(s(z0)) → s(proper(z0)) 164.50/58.79
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 164.50/58.79
proper(0) → ok(0) 164.50/58.79
top(mark(z0)) → top(proper(z0)) 164.50/58.79
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2)) 164.50/58.79
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0)) 164.50/58.79
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1)) 164.50/58.79
FROM(mark(z0)) → c8(FROM(z0)) 164.50/58.79
FROM(ok(z0)) → c9(FROM(z0)) 164.50/58.79
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.50/58.79
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.50/58.79
S(mark(z0)) → c12(S(z0)) 164.50/58.79
S(ok(z0)) → c13(S(z0)) 164.50/58.79
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.50/58.79
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.50/58.79
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1)) 164.50/58.79
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 164.50/58.79
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0)) 164.50/58.79
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 164.50/58.79
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 164.50/58.79
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 164.50/58.79
ACTIVE(from(z0)) → c(S(z0)) 164.50/58.79
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.50/58.79
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.50/58.79
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.50/58.79
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 164.50/58.79
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.50/58.79
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 164.50/58.79
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.50/58.79
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.50/58.79
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 164.50/58.79
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 164.50/58.79
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.50/58.79
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 164.50/58.79
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 164.50/58.79
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 164.50/58.79
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 164.50/58.79
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 164.50/58.79
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.50/58.79
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.50/58.79
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.50/58.79
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 164.50/58.79
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.50/58.79
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 164.50/58.79
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.50/58.79
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.50/58.79
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 164.50/58.79
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.50/58.79
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 164.50/58.79
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.50/58.79
PROPER(from(0)) → c17(FROM(ok(0)), PROPER(0))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2)) 164.50/58.79
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0)) 164.50/58.79
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1)) 164.50/58.79
FROM(mark(z0)) → c8(FROM(z0)) 164.50/58.79
FROM(ok(z0)) → c9(FROM(z0)) 164.50/58.79
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.50/58.79
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.50/58.79
S(mark(z0)) → c12(S(z0)) 164.50/58.79
S(ok(z0)) → c13(S(z0)) 164.50/58.79
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.50/58.79
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.50/58.79
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1)) 164.50/58.79
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 164.50/58.79
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0)) 164.50/58.79
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 164.50/58.79
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 164.50/58.79
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 164.50/58.79
ACTIVE(from(z0)) → c(S(z0)) 164.50/58.79
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.50/58.79
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.50/58.79
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.50/58.79
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 164.50/58.79
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.50/58.79
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 164.50/58.79
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.50/58.79
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.50/58.79
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 164.50/58.79
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 164.50/58.79
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.50/58.79
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 164.50/58.79
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 164.50/58.79
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 164.50/58.79
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 164.50/58.79
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 164.50/58.79
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.50/58.79
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.50/58.79
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.50/58.79
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 164.50/58.79
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.50/58.79
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 164.50/58.79
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.50/58.79
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.50/58.79
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 164.50/58.79
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.50/58.79
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 164.50/58.79
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.50/58.79
PROPER(from(0)) → c17(FROM(ok(0)), PROPER(0))

K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, PROPER, TOP

Compound Symbols:

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

164.50/58.79

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

Removed 1 trailing tuple parts

164.50/58.79

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 164.50/58.79
active(sel(0, cons(z0, z1))) → mark(z0) 164.50/58.79
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 164.50/58.79
active(from(z0)) → from(active(z0)) 164.50/58.79
active(cons(z0, z1)) → cons(active(z0), z1) 164.50/58.79
active(s(z0)) → s(active(z0)) 164.50/58.79
active(sel(z0, z1)) → sel(active(z0), z1) 164.50/58.79
active(sel(z0, z1)) → sel(z0, active(z1)) 164.50/58.79
from(mark(z0)) → mark(from(z0)) 164.50/58.79
from(ok(z0)) → ok(from(z0)) 164.50/58.79
cons(mark(z0), z1) → mark(cons(z0, z1)) 164.50/58.79
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 164.50/58.79
s(mark(z0)) → mark(s(z0)) 164.50/58.79
s(ok(z0)) → ok(s(z0)) 164.50/58.79
sel(mark(z0), z1) → mark(sel(z0, z1)) 164.50/58.79
sel(z0, mark(z1)) → mark(sel(z0, z1)) 164.50/58.79
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 164.50/58.79
proper(from(z0)) → from(proper(z0)) 164.50/58.79
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 164.50/58.79
proper(s(z0)) → s(proper(z0)) 164.50/58.79
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 164.50/58.79
proper(0) → ok(0) 164.50/58.79
top(mark(z0)) → top(proper(z0)) 164.50/58.79
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2)) 164.50/58.79
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0)) 164.50/58.79
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1)) 164.50/58.79
FROM(mark(z0)) → c8(FROM(z0)) 164.50/58.79
FROM(ok(z0)) → c9(FROM(z0)) 164.50/58.79
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.50/58.79
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.50/58.79
S(mark(z0)) → c12(S(z0)) 164.50/58.79
S(ok(z0)) → c13(S(z0)) 164.50/58.79
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.50/58.79
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.50/58.79
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1)) 164.50/58.79
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 164.50/58.79
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0)) 164.50/58.79
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 164.50/58.79
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 164.50/58.79
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 164.50/58.79
ACTIVE(from(z0)) → c(S(z0)) 164.50/58.79
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.50/58.79
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.50/58.79
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.50/58.79
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 164.50/58.79
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.50/58.79
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 164.50/58.79
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.50/58.79
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.50/58.79
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 164.50/58.79
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 164.50/58.79
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.50/58.79
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 164.50/58.79
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 164.50/58.79
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 164.50/58.79
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 164.50/58.79
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 164.50/58.79
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.50/58.79
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.50/58.79
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.50/58.79
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 164.50/58.79
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.50/58.79
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 164.50/58.79
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.50/58.79
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.50/58.79
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 164.50/58.79
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.50/58.79
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 164.50/58.79
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.50/58.79
PROPER(from(0)) → c17(FROM(ok(0)))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2)) 164.50/58.79
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0)) 164.50/58.79
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1)) 164.50/58.79
FROM(mark(z0)) → c8(FROM(z0)) 164.50/58.79
FROM(ok(z0)) → c9(FROM(z0)) 164.50/58.79
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.50/58.79
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.50/58.79
S(mark(z0)) → c12(S(z0)) 164.50/58.79
S(ok(z0)) → c13(S(z0)) 164.50/58.79
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.50/58.79
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.50/58.79
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1)) 164.50/58.79
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 164.50/58.79
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0)) 164.50/58.79
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 164.50/58.79
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 164.50/58.79
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 164.50/58.79
ACTIVE(from(z0)) → c(S(z0)) 164.50/58.79
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.50/58.79
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.50/58.79
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.50/58.79
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 164.50/58.79
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.50/58.79
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 164.50/58.79
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.50/58.79
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.50/58.79
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 164.50/58.79
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 164.50/58.79
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.50/58.79
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 164.50/58.79
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 164.50/58.79
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 164.50/58.79
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 164.50/58.79
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 164.50/58.79
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.50/58.79
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.50/58.79
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.50/58.79
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 164.50/58.79
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.50/58.79
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 164.50/58.79
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.50/58.79
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.50/58.79
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 164.50/58.79
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.83
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 164.91/58.83
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.83
PROPER(from(0)) → c17(FROM(ok(0)))

K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, PROPER, TOP

Compound Symbols:

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

164.91/58.83

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

Use narrowing to replace PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 164.91/58.83
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 164.91/58.83
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 164.91/58.83
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 164.91/58.83
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 164.91/58.83
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 164.91/58.83
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 164.91/58.83
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 164.91/58.83
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 164.91/58.83
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))

164.91/58.83

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 164.91/58.83
active(sel(0, cons(z0, z1))) → mark(z0) 164.91/58.83
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 164.91/58.83
active(from(z0)) → from(active(z0)) 164.91/58.83
active(cons(z0, z1)) → cons(active(z0), z1) 164.91/58.83
active(s(z0)) → s(active(z0)) 164.91/58.83
active(sel(z0, z1)) → sel(active(z0), z1) 164.91/58.83
active(sel(z0, z1)) → sel(z0, active(z1)) 164.91/58.83
from(mark(z0)) → mark(from(z0)) 164.91/58.83
from(ok(z0)) → ok(from(z0)) 164.91/58.83
cons(mark(z0), z1) → mark(cons(z0, z1)) 164.91/58.83
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 164.91/58.83
s(mark(z0)) → mark(s(z0)) 164.91/58.83
s(ok(z0)) → ok(s(z0)) 164.91/58.83
sel(mark(z0), z1) → mark(sel(z0, z1)) 164.91/58.83
sel(z0, mark(z1)) → mark(sel(z0, z1)) 164.91/58.83
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 164.91/58.83
proper(from(z0)) → from(proper(z0)) 164.91/58.83
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 164.91/58.83
proper(s(z0)) → s(proper(z0)) 164.91/58.83
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 164.91/58.83
proper(0) → ok(0) 164.91/58.83
top(mark(z0)) → top(proper(z0)) 164.91/58.83
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2)) 164.91/58.83
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0)) 164.91/58.83
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1)) 164.91/58.83
FROM(mark(z0)) → c8(FROM(z0)) 164.91/58.83
FROM(ok(z0)) → c9(FROM(z0)) 164.91/58.83
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.91/58.83
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.91/58.83
S(mark(z0)) → c12(S(z0)) 164.91/58.83
S(ok(z0)) → c13(S(z0)) 164.91/58.83
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.91/58.83
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.91/58.83
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1)) 164.91/58.83
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0)) 164.91/58.83
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 164.91/58.83
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 164.91/58.83
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 164.91/58.83
ACTIVE(from(z0)) → c(S(z0)) 164.91/58.83
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.83
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.83
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.83
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 164.91/58.83
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.83
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 164.91/58.83
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.83
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.83
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 164.91/58.83
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.83
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.83
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 164.91/58.83
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 164.91/58.83
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 164.91/58.83
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 164.91/58.83
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 164.91/58.83
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.83
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.83
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.83
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 164.91/58.83
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.83
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 164.91/58.83
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.83
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.83
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 164.91/58.83
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.83
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 164.91/58.83
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.83
PROPER(from(0)) → c17(FROM(ok(0))) 164.91/58.83
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 164.91/58.83
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 164.91/58.83
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 164.91/58.83
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 164.91/58.83
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 164.91/58.83
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 164.91/58.83
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 164.91/58.83
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 164.91/58.83
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 164.91/58.83
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2)) 164.91/58.83
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0)) 164.91/58.83
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1)) 164.91/58.83
FROM(mark(z0)) → c8(FROM(z0)) 164.91/58.83
FROM(ok(z0)) → c9(FROM(z0)) 164.91/58.83
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.91/58.83
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.91/58.83
S(mark(z0)) → c12(S(z0)) 164.91/58.83
S(ok(z0)) → c13(S(z0)) 164.91/58.83
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.91/58.83
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.91/58.83
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1)) 164.91/58.83
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0)) 164.91/58.83
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 164.91/58.83
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 164.91/58.83
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 164.91/58.83
ACTIVE(from(z0)) → c(S(z0)) 164.91/58.83
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.83
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.83
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.83
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 164.91/58.83
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.83
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 164.91/58.83
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.83
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.83
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 164.91/58.83
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.83
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.83
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 164.91/58.83
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 164.91/58.83
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 164.91/58.83
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 164.91/58.83
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 164.91/58.84
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.84
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.84
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.84
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 164.91/58.84
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.84
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 164.91/58.84
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.84
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.84
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 164.91/58.84
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.84
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 164.91/58.84
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.84
PROPER(from(0)) → c17(FROM(ok(0))) 164.91/58.84
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 164.91/58.84
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 164.91/58.84
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 164.91/58.84
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 164.91/58.84
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 164.91/58.84
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 164.91/58.84
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 164.91/58.84
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 164.91/58.84
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 164.91/58.84
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, PROPER, TOP

Compound Symbols:

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

164.91/58.84

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

Removed 2 trailing tuple parts

164.91/58.84

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 164.91/58.84
active(sel(0, cons(z0, z1))) → mark(z0) 164.91/58.84
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 164.91/58.84
active(from(z0)) → from(active(z0)) 164.91/58.84
active(cons(z0, z1)) → cons(active(z0), z1) 164.91/58.84
active(s(z0)) → s(active(z0)) 164.91/58.84
active(sel(z0, z1)) → sel(active(z0), z1) 164.91/58.84
active(sel(z0, z1)) → sel(z0, active(z1)) 164.91/58.84
from(mark(z0)) → mark(from(z0)) 164.91/58.84
from(ok(z0)) → ok(from(z0)) 164.91/58.84
cons(mark(z0), z1) → mark(cons(z0, z1)) 164.91/58.84
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 164.91/58.84
s(mark(z0)) → mark(s(z0)) 164.91/58.84
s(ok(z0)) → ok(s(z0)) 164.91/58.84
sel(mark(z0), z1) → mark(sel(z0, z1)) 164.91/58.84
sel(z0, mark(z1)) → mark(sel(z0, z1)) 164.91/58.84
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 164.91/58.84
proper(from(z0)) → from(proper(z0)) 164.91/58.84
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 164.91/58.84
proper(s(z0)) → s(proper(z0)) 164.91/58.84
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 164.91/58.84
proper(0) → ok(0) 164.91/58.84
top(mark(z0)) → top(proper(z0)) 164.91/58.84
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2)) 164.91/58.84
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0)) 164.91/58.84
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1)) 164.91/58.84
FROM(mark(z0)) → c8(FROM(z0)) 164.91/58.84
FROM(ok(z0)) → c9(FROM(z0)) 164.91/58.84
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.91/58.84
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.91/58.84
S(mark(z0)) → c12(S(z0)) 164.91/58.84
S(ok(z0)) → c13(S(z0)) 164.91/58.84
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.91/58.84
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.91/58.84
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1)) 164.91/58.84
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0)) 164.91/58.84
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 164.91/58.84
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 164.91/58.84
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 164.91/58.84
ACTIVE(from(z0)) → c(S(z0)) 164.91/58.84
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.84
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.84
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.84
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 164.91/58.84
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.84
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 164.91/58.84
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.84
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.84
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 164.91/58.84
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.84
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.84
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 164.91/58.84
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 164.91/58.84
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 164.91/58.84
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 164.91/58.84
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 164.91/58.84
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.84
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.84
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.84
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 164.91/58.84
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.84
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 164.91/58.84
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.84
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.84
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 164.91/58.84
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.84
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 164.91/58.84
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.84
PROPER(from(0)) → c17(FROM(ok(0))) 164.91/58.84
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 164.91/58.84
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 164.91/58.84
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 164.91/58.84
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 164.91/58.84
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 164.91/58.84
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 164.91/58.84
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 164.91/58.84
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 164.91/58.84
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0)) 164.91/58.84
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2)) 164.91/58.84
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0)) 164.91/58.84
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1)) 164.91/58.84
FROM(mark(z0)) → c8(FROM(z0)) 164.91/58.84
FROM(ok(z0)) → c9(FROM(z0)) 164.91/58.84
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.91/58.84
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.91/58.84
S(mark(z0)) → c12(S(z0)) 164.91/58.84
S(ok(z0)) → c13(S(z0)) 164.91/58.84
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.91/58.84
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.91/58.84
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1)) 164.91/58.84
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0)) 164.91/58.84
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 164.91/58.84
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 164.91/58.84
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 164.91/58.84
ACTIVE(from(z0)) → c(S(z0)) 164.91/58.84
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.84
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.84
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.84
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 164.91/58.84
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.84
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 164.91/58.84
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.84
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.84
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 164.91/58.84
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.84
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.84
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 164.91/58.84
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 164.91/58.84
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 164.91/58.84
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 164.91/58.84
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 164.91/58.84
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.84
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.84
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.84
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 164.91/58.84
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.84
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 164.91/58.84
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.84
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.84
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 164.91/58.84
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.84
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 164.91/58.84
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.84
PROPER(from(0)) → c17(FROM(ok(0))) 164.91/58.84
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 164.91/58.84
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 164.91/58.84
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 164.91/58.84
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 164.91/58.84
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 164.91/58.84
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 164.91/58.84
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 164.91/58.84
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 164.91/58.84
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0)) 164.91/58.84
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, PROPER, TOP

Compound Symbols:

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

164.91/58.84

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

Use narrowing to replace PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0)) by

PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0))) 164.91/58.84
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.84
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0))) 164.91/58.84
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.84
PROPER(s(0)) → c19(S(ok(0)), PROPER(0))

164.91/58.84

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 164.91/58.84
active(sel(0, cons(z0, z1))) → mark(z0) 164.91/58.84
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 164.91/58.84
active(from(z0)) → from(active(z0)) 164.91/58.84
active(cons(z0, z1)) → cons(active(z0), z1) 164.91/58.84
active(s(z0)) → s(active(z0)) 164.91/58.84
active(sel(z0, z1)) → sel(active(z0), z1) 164.91/58.84
active(sel(z0, z1)) → sel(z0, active(z1)) 164.91/58.84
from(mark(z0)) → mark(from(z0)) 164.91/58.84
from(ok(z0)) → ok(from(z0)) 164.91/58.84
cons(mark(z0), z1) → mark(cons(z0, z1)) 164.91/58.84
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 164.91/58.84
s(mark(z0)) → mark(s(z0)) 164.91/58.84
s(ok(z0)) → ok(s(z0)) 164.91/58.84
sel(mark(z0), z1) → mark(sel(z0, z1)) 164.91/58.84
sel(z0, mark(z1)) → mark(sel(z0, z1)) 164.91/58.84
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 164.91/58.84
proper(from(z0)) → from(proper(z0)) 164.91/58.84
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 164.91/58.84
proper(s(z0)) → s(proper(z0)) 164.91/58.84
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 164.91/58.84
proper(0) → ok(0) 164.91/58.84
top(mark(z0)) → top(proper(z0)) 164.91/58.84
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2)) 164.91/58.84
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0)) 164.91/58.84
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1)) 164.91/58.84
FROM(mark(z0)) → c8(FROM(z0)) 164.91/58.84
FROM(ok(z0)) → c9(FROM(z0)) 164.91/58.84
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.91/58.84
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.91/58.84
S(mark(z0)) → c12(S(z0)) 164.91/58.84
S(ok(z0)) → c13(S(z0)) 164.91/58.84
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.91/58.84
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.91/58.84
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1)) 164.91/58.84
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 164.91/58.84
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 164.91/58.84
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 164.91/58.84
ACTIVE(from(z0)) → c(S(z0)) 164.91/58.84
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.84
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.84
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.84
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 164.91/58.84
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.84
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 164.91/58.84
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.84
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.84
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 164.91/58.84
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.84
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.84
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 164.91/58.84
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 164.91/58.84
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 164.91/58.84
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 164.91/58.84
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 164.91/58.84
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.84
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.84
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.84
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 164.91/58.84
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.84
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 164.91/58.84
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.84
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.84
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 164.91/58.84
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.84
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 164.91/58.84
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.84
PROPER(from(0)) → c17(FROM(ok(0))) 164.91/58.84
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 164.91/58.84
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 164.91/58.84
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 164.91/58.84
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 164.91/58.84
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 164.91/58.84
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 164.91/58.84
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 164.91/58.84
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 164.91/58.84
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0)) 164.91/58.84
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1)) 164.91/58.84
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0))) 164.91/58.84
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.84
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0))) 164.91/58.84
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.84
PROPER(s(0)) → c19(S(ok(0)), PROPER(0))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2)) 164.91/58.84
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0)) 164.91/58.84
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1)) 164.91/58.84
FROM(mark(z0)) → c8(FROM(z0)) 164.91/58.84
FROM(ok(z0)) → c9(FROM(z0)) 164.91/58.84
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.91/58.84
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.91/58.84
S(mark(z0)) → c12(S(z0)) 164.91/58.84
S(ok(z0)) → c13(S(z0)) 164.91/58.84
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.91/58.84
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.91/58.84
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1)) 164.91/58.84
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 164.91/58.84
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 164.91/58.84
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 164.91/58.84
ACTIVE(from(z0)) → c(S(z0)) 164.91/58.84
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.84
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.84
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.84
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 164.91/58.84
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.84
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 164.91/58.84
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.84
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.84
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 164.91/58.84
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.84
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.84
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 164.91/58.84
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 164.91/58.84
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 164.91/58.84
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 164.91/58.84
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 164.91/58.84
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.84
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.84
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.84
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 164.91/58.84
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.84
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 164.91/58.84
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.85
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.85
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 164.91/58.85
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.85
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 164.91/58.85
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.85
PROPER(from(0)) → c17(FROM(ok(0))) 164.91/58.85
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 164.91/58.85
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 164.91/58.85
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 164.91/58.85
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 164.91/58.85
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 164.91/58.85
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 164.91/58.85
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 164.91/58.85
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 164.91/58.85
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0)) 164.91/58.85
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1)) 164.91/58.85
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0))) 164.91/58.85
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.85
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0))) 164.91/58.85
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.85
PROPER(s(0)) → c19(S(ok(0)), PROPER(0))

K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, PROPER, TOP

Compound Symbols:

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

164.91/58.85

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

Removed 1 trailing tuple parts

164.91/58.85

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 164.91/58.85
active(sel(0, cons(z0, z1))) → mark(z0) 164.91/58.85
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 164.91/58.85
active(from(z0)) → from(active(z0)) 164.91/58.85
active(cons(z0, z1)) → cons(active(z0), z1) 164.91/58.85
active(s(z0)) → s(active(z0)) 164.91/58.85
active(sel(z0, z1)) → sel(active(z0), z1) 164.91/58.85
active(sel(z0, z1)) → sel(z0, active(z1)) 164.91/58.85
from(mark(z0)) → mark(from(z0)) 164.91/58.85
from(ok(z0)) → ok(from(z0)) 164.91/58.85
cons(mark(z0), z1) → mark(cons(z0, z1)) 164.91/58.85
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 164.91/58.85
s(mark(z0)) → mark(s(z0)) 164.91/58.85
s(ok(z0)) → ok(s(z0)) 164.91/58.85
sel(mark(z0), z1) → mark(sel(z0, z1)) 164.91/58.85
sel(z0, mark(z1)) → mark(sel(z0, z1)) 164.91/58.85
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 164.91/58.85
proper(from(z0)) → from(proper(z0)) 164.91/58.85
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 164.91/58.85
proper(s(z0)) → s(proper(z0)) 164.91/58.85
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 164.91/58.85
proper(0) → ok(0) 164.91/58.85
top(mark(z0)) → top(proper(z0)) 164.91/58.85
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2)) 164.91/58.85
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0)) 164.91/58.85
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1)) 164.91/58.85
FROM(mark(z0)) → c8(FROM(z0)) 164.91/58.85
FROM(ok(z0)) → c9(FROM(z0)) 164.91/58.85
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.91/58.85
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.91/58.85
S(mark(z0)) → c12(S(z0)) 164.91/58.85
S(ok(z0)) → c13(S(z0)) 164.91/58.85
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.91/58.85
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.91/58.85
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1)) 164.91/58.85
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 164.91/58.85
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 164.91/58.85
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 164.91/58.85
ACTIVE(from(z0)) → c(S(z0)) 164.91/58.85
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.85
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.85
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.85
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 164.91/58.85
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.85
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 164.91/58.85
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.85
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.85
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 164.91/58.85
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.85
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.85
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 164.91/58.85
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 164.91/58.85
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 164.91/58.85
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 164.91/58.85
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 164.91/58.85
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.85
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.85
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.85
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 164.91/58.85
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.85
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 164.91/58.85
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.85
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.85
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 164.91/58.85
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.85
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 164.91/58.85
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.85
PROPER(from(0)) → c17(FROM(ok(0))) 164.91/58.85
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 164.91/58.85
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 164.91/58.85
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 164.91/58.85
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 164.91/58.85
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 164.91/58.85
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 164.91/58.85
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 164.91/58.85
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 164.91/58.85
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0)) 164.91/58.85
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1)) 164.91/58.85
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0))) 164.91/58.85
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.85
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0))) 164.91/58.85
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.85
PROPER(s(0)) → c19(S(ok(0)))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2)) 164.91/58.85
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0)) 164.91/58.85
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1)) 164.91/58.85
FROM(mark(z0)) → c8(FROM(z0)) 164.91/58.85
FROM(ok(z0)) → c9(FROM(z0)) 164.91/58.85
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.91/58.85
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.91/58.85
S(mark(z0)) → c12(S(z0)) 164.91/58.85
S(ok(z0)) → c13(S(z0)) 164.91/58.85
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.91/58.85
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.91/58.85
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1)) 164.91/58.85
PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 164.91/58.85
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 164.91/58.85
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 164.91/58.85
ACTIVE(from(z0)) → c(S(z0)) 164.91/58.85
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.85
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.85
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.85
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 164.91/58.85
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.85
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 164.91/58.85
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.85
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.85
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 164.91/58.85
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.85
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.85
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 164.91/58.85
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 164.91/58.85
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 164.91/58.85
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 164.91/58.85
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 164.91/58.85
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.85
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.85
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.85
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 164.91/58.85
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.85
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 164.91/58.85
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.85
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.85
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 164.91/58.85
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.85
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 164.91/58.85
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.85
PROPER(from(0)) → c17(FROM(ok(0))) 164.91/58.85
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 164.91/58.85
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 164.91/58.85
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 164.91/58.85
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 164.91/58.85
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 164.91/58.85
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 164.91/58.85
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 164.91/58.85
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 164.91/58.86
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0)) 164.91/58.86
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1)) 164.91/58.86
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0))) 164.91/58.86
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.86
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0))) 164.91/58.86
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.86
PROPER(s(0)) → c19(S(ok(0)))

K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, PROPER, TOP

Compound Symbols:

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

164.91/58.86

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

Use narrowing to replace PROPER(sel(z0, z1)) → c20(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(sel(x0, from(z0))) → c20(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 164.91/58.86
PROPER(sel(x0, cons(z0, z1))) → c20(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 164.91/58.86
PROPER(sel(x0, s(z0))) → c20(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 164.91/58.86
PROPER(sel(x0, sel(z0, z1))) → c20(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 164.91/58.86
PROPER(sel(x0, 0)) → c20(SEL(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 164.91/58.86
PROPER(sel(from(z0), x1)) → c20(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 164.91/58.86
PROPER(sel(cons(z0, z1), x1)) → c20(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 164.91/58.86
PROPER(sel(s(z0), x1)) → c20(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 164.91/58.86
PROPER(sel(sel(z0, z1), x1)) → c20(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 164.91/58.86
PROPER(sel(0, x1)) → c20(SEL(ok(0), proper(x1)), PROPER(0), PROPER(x1))

164.91/58.86

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 164.91/58.86
active(sel(0, cons(z0, z1))) → mark(z0) 164.91/58.86
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 164.91/58.86
active(from(z0)) → from(active(z0)) 164.91/58.86
active(cons(z0, z1)) → cons(active(z0), z1) 164.91/58.86
active(s(z0)) → s(active(z0)) 164.91/58.86
active(sel(z0, z1)) → sel(active(z0), z1) 164.91/58.86
active(sel(z0, z1)) → sel(z0, active(z1)) 164.91/58.86
from(mark(z0)) → mark(from(z0)) 164.91/58.86
from(ok(z0)) → ok(from(z0)) 164.91/58.86
cons(mark(z0), z1) → mark(cons(z0, z1)) 164.91/58.86
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 164.91/58.86
s(mark(z0)) → mark(s(z0)) 164.91/58.86
s(ok(z0)) → ok(s(z0)) 164.91/58.86
sel(mark(z0), z1) → mark(sel(z0, z1)) 164.91/58.86
sel(z0, mark(z1)) → mark(sel(z0, z1)) 164.91/58.86
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 164.91/58.86
proper(from(z0)) → from(proper(z0)) 164.91/58.86
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 164.91/58.86
proper(s(z0)) → s(proper(z0)) 164.91/58.86
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 164.91/58.86
proper(0) → ok(0) 164.91/58.86
top(mark(z0)) → top(proper(z0)) 164.91/58.86
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2)) 164.91/58.86
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0)) 164.91/58.86
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1)) 164.91/58.86
FROM(mark(z0)) → c8(FROM(z0)) 164.91/58.86
FROM(ok(z0)) → c9(FROM(z0)) 164.91/58.86
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.91/58.86
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.91/58.86
S(mark(z0)) → c12(S(z0)) 164.91/58.86
S(ok(z0)) → c13(S(z0)) 164.91/58.86
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.91/58.86
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.91/58.86
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1)) 164.91/58.86
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 164.91/58.86
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 164.91/58.86
ACTIVE(from(z0)) → c(S(z0)) 164.91/58.86
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.86
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.86
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.86
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 164.91/58.86
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.86
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 164.91/58.86
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.86
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.86
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 164.91/58.86
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.86
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.86
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 164.91/58.86
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 164.91/58.86
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 164.91/58.86
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 164.91/58.86
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 164.91/58.86
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.86
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.86
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.86
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 164.91/58.86
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.86
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 164.91/58.86
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.86
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.86
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 164.91/58.86
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.86
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 164.91/58.86
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.86
PROPER(from(0)) → c17(FROM(ok(0))) 164.91/58.86
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 164.91/58.86
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 164.91/58.86
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 164.91/58.86
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 164.91/58.86
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 164.91/58.86
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 164.91/58.86
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 164.91/58.86
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 164.91/58.86
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0)) 164.91/58.86
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1)) 164.91/58.86
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0))) 164.91/58.86
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.86
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0))) 164.91/58.86
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.86
PROPER(s(0)) → c19(S(ok(0))) 164.91/58.86
PROPER(sel(x0, from(z0))) → c20(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 164.91/58.86
PROPER(sel(x0, cons(z0, z1))) → c20(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 164.91/58.86
PROPER(sel(x0, s(z0))) → c20(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 164.91/58.86
PROPER(sel(x0, sel(z0, z1))) → c20(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 164.91/58.86
PROPER(sel(x0, 0)) → c20(SEL(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 164.91/58.86
PROPER(sel(from(z0), x1)) → c20(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 164.91/58.86
PROPER(sel(cons(z0, z1), x1)) → c20(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 164.91/58.86
PROPER(sel(s(z0), x1)) → c20(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 164.91/58.86
PROPER(sel(sel(z0, z1), x1)) → c20(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 164.91/58.86
PROPER(sel(0, x1)) → c20(SEL(ok(0), proper(x1)), PROPER(0), PROPER(x1))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2)) 164.91/58.86
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0)) 164.91/58.86
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1)) 164.91/58.86
FROM(mark(z0)) → c8(FROM(z0)) 164.91/58.86
FROM(ok(z0)) → c9(FROM(z0)) 164.91/58.86
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.91/58.86
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.91/58.86
S(mark(z0)) → c12(S(z0)) 164.91/58.86
S(ok(z0)) → c13(S(z0)) 164.91/58.86
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.91/58.86
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.91/58.86
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1)) 164.91/58.86
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 164.91/58.86
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 164.91/58.86
ACTIVE(from(z0)) → c(S(z0)) 164.91/58.86
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.86
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.86
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.86
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 164.91/58.86
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.86
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 164.91/58.86
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.86
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.86
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 164.91/58.86
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.86
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.86
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 164.91/58.86
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 164.91/58.86
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 164.91/58.86
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 164.91/58.86
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 164.91/58.86
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.86
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.86
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.86
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 164.91/58.86
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.86
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 164.91/58.86
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.86
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.86
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 164.91/58.86
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.86
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 164.91/58.86
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.86
PROPER(from(0)) → c17(FROM(ok(0))) 164.91/58.86
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 164.91/58.86
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 164.91/58.89
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 164.91/58.89
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 164.91/58.89
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 164.91/58.89
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 164.91/58.89
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 164.91/58.89
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 164.91/58.89
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0)) 164.91/58.89
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1)) 164.91/58.89
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0))) 164.91/58.89
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.89
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0))) 164.91/58.89
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.89
PROPER(s(0)) → c19(S(ok(0))) 164.91/58.89
PROPER(sel(x0, from(z0))) → c20(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 164.91/58.89
PROPER(sel(x0, cons(z0, z1))) → c20(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 164.91/58.89
PROPER(sel(x0, s(z0))) → c20(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 164.91/58.89
PROPER(sel(x0, sel(z0, z1))) → c20(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 164.91/58.89
PROPER(sel(x0, 0)) → c20(SEL(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 164.91/58.89
PROPER(sel(from(z0), x1)) → c20(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 164.91/58.89
PROPER(sel(cons(z0, z1), x1)) → c20(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 164.91/58.89
PROPER(sel(s(z0), x1)) → c20(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 164.91/58.89
PROPER(sel(sel(z0, z1), x1)) → c20(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 164.91/58.89
PROPER(sel(0, x1)) → c20(SEL(ok(0), proper(x1)), PROPER(0), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, TOP, PROPER

Compound Symbols:

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

164.91/58.89

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

Removed 2 trailing tuple parts

164.91/58.89

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 164.91/58.89
active(sel(0, cons(z0, z1))) → mark(z0) 164.91/58.89
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 164.91/58.89
active(from(z0)) → from(active(z0)) 164.91/58.89
active(cons(z0, z1)) → cons(active(z0), z1) 164.91/58.89
active(s(z0)) → s(active(z0)) 164.91/58.89
active(sel(z0, z1)) → sel(active(z0), z1) 164.91/58.89
active(sel(z0, z1)) → sel(z0, active(z1)) 164.91/58.89
from(mark(z0)) → mark(from(z0)) 164.91/58.89
from(ok(z0)) → ok(from(z0)) 164.91/58.89
cons(mark(z0), z1) → mark(cons(z0, z1)) 164.91/58.89
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 164.91/58.89
s(mark(z0)) → mark(s(z0)) 164.91/58.89
s(ok(z0)) → ok(s(z0)) 164.91/58.89
sel(mark(z0), z1) → mark(sel(z0, z1)) 164.91/58.89
sel(z0, mark(z1)) → mark(sel(z0, z1)) 164.91/58.89
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 164.91/58.89
proper(from(z0)) → from(proper(z0)) 164.91/58.89
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 164.91/58.89
proper(s(z0)) → s(proper(z0)) 164.91/58.89
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 164.91/58.89
proper(0) → ok(0) 164.91/58.89
top(mark(z0)) → top(proper(z0)) 164.91/58.89
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2)) 164.91/58.89
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0)) 164.91/58.89
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1)) 164.91/58.89
FROM(mark(z0)) → c8(FROM(z0)) 164.91/58.89
FROM(ok(z0)) → c9(FROM(z0)) 164.91/58.89
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.91/58.89
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.91/58.89
S(mark(z0)) → c12(S(z0)) 164.91/58.89
S(ok(z0)) → c13(S(z0)) 164.91/58.89
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.91/58.89
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.91/58.89
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1)) 164.91/58.89
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 164.91/58.89
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 164.91/58.89
ACTIVE(from(z0)) → c(S(z0)) 164.91/58.89
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.89
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.89
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.89
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 164.91/58.89
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.89
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 164.91/58.89
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.89
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.89
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 164.91/58.89
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.89
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.89
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 164.91/58.89
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 164.91/58.89
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 164.91/58.89
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 164.91/58.89
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 164.91/58.89
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.89
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.89
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.89
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 164.91/58.89
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.89
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 164.91/58.89
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.89
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.89
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 164.91/58.89
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.89
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 164.91/58.89
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.89
PROPER(from(0)) → c17(FROM(ok(0))) 164.91/58.89
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 164.91/58.89
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 164.91/58.89
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 164.91/58.89
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 164.91/58.89
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 164.91/58.89
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 164.91/58.89
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 164.91/58.89
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 164.91/58.89
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0)) 164.91/58.89
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1)) 164.91/58.89
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0))) 164.91/58.89
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.89
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0))) 164.91/58.89
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.89
PROPER(s(0)) → c19(S(ok(0))) 164.91/58.89
PROPER(sel(x0, from(z0))) → c20(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 164.91/58.89
PROPER(sel(x0, cons(z0, z1))) → c20(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 164.91/58.89
PROPER(sel(x0, s(z0))) → c20(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 164.91/58.89
PROPER(sel(x0, sel(z0, z1))) → c20(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 164.91/58.89
PROPER(sel(from(z0), x1)) → c20(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 164.91/58.89
PROPER(sel(cons(z0, z1), x1)) → c20(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 164.91/58.89
PROPER(sel(s(z0), x1)) → c20(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 164.91/58.89
PROPER(sel(sel(z0, z1), x1)) → c20(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 164.91/58.89
PROPER(sel(x0, 0)) → c20(SEL(proper(x0), ok(0)), PROPER(x0)) 164.91/58.89
PROPER(sel(0, x1)) → c20(SEL(ok(0), proper(x1)), PROPER(x1))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2)) 164.91/58.89
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0)) 164.91/58.89
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1)) 164.91/58.89
FROM(mark(z0)) → c8(FROM(z0)) 164.91/58.89
FROM(ok(z0)) → c9(FROM(z0)) 164.91/58.89
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.91/58.89
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.91/58.89
S(mark(z0)) → c12(S(z0)) 164.91/58.89
S(ok(z0)) → c13(S(z0)) 164.91/58.89
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.91/58.89
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.91/58.89
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1)) 164.91/58.89
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 164.91/58.89
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 164.91/58.89
ACTIVE(from(z0)) → c(S(z0)) 164.91/58.89
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.89
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.89
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.89
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 164.91/58.89
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.89
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 164.91/58.89
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.89
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.89
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 164.91/58.89
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.90
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.90
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 164.91/58.90
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 164.91/58.90
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.90
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.90
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.90
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 164.91/58.90
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.90
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 164.91/58.90
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.90
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 164.91/58.90
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.90
PROPER(from(0)) → c17(FROM(ok(0))) 164.91/58.90
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 164.91/58.90
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 164.91/58.90
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 164.91/58.90
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 164.91/58.90
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 164.91/58.90
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 164.91/58.90
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 164.91/58.90
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 164.91/58.90
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0)) 164.91/58.90
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1)) 164.91/58.90
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0))) 164.91/58.90
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.90
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0))) 164.91/58.90
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.90
PROPER(s(0)) → c19(S(ok(0))) 164.91/58.90
PROPER(sel(x0, from(z0))) → c20(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 164.91/58.90
PROPER(sel(x0, cons(z0, z1))) → c20(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 164.91/58.90
PROPER(sel(x0, s(z0))) → c20(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 164.91/58.90
PROPER(sel(x0, sel(z0, z1))) → c20(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 164.91/58.90
PROPER(sel(from(z0), x1)) → c20(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 164.91/58.90
PROPER(sel(cons(z0, z1), x1)) → c20(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 164.91/58.90
PROPER(sel(s(z0), x1)) → c20(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 164.91/58.90
PROPER(sel(sel(z0, z1), x1)) → c20(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 164.91/58.90
PROPER(sel(x0, 0)) → c20(SEL(proper(x0), ok(0)), PROPER(x0)) 164.91/58.90
PROPER(sel(0, x1)) → c20(SEL(ok(0), proper(x1)), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, TOP, PROPER

Compound Symbols:

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

164.91/58.90

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

Use narrowing to replace TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0))) 164.91/58.90
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.90
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0))) 164.91/58.90
TOP(mark(sel(z0, z1))) → c22(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.90
TOP(mark(0)) → c22(TOP(ok(0)), PROPER(0))

164.91/58.90

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 164.91/58.90
active(sel(0, cons(z0, z1))) → mark(z0) 164.91/58.90
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 164.91/58.90
active(from(z0)) → from(active(z0)) 164.91/58.90
active(cons(z0, z1)) → cons(active(z0), z1) 164.91/58.90
active(s(z0)) → s(active(z0)) 164.91/58.90
active(sel(z0, z1)) → sel(active(z0), z1) 164.91/58.90
active(sel(z0, z1)) → sel(z0, active(z1)) 164.91/58.90
from(mark(z0)) → mark(from(z0)) 164.91/58.90
from(ok(z0)) → ok(from(z0)) 164.91/58.90
cons(mark(z0), z1) → mark(cons(z0, z1)) 164.91/58.90
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 164.91/58.90
s(mark(z0)) → mark(s(z0)) 164.91/58.90
s(ok(z0)) → ok(s(z0)) 164.91/58.90
sel(mark(z0), z1) → mark(sel(z0, z1)) 164.91/58.90
sel(z0, mark(z1)) → mark(sel(z0, z1)) 164.91/58.90
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 164.91/58.90
proper(from(z0)) → from(proper(z0)) 164.91/58.90
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 164.91/58.90
proper(s(z0)) → s(proper(z0)) 164.91/58.90
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 164.91/58.90
proper(0) → ok(0) 164.91/58.90
top(mark(z0)) → top(proper(z0)) 164.91/58.90
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2)) 164.91/58.90
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0)) 164.91/58.90
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1)) 164.91/58.90
FROM(mark(z0)) → c8(FROM(z0)) 164.91/58.90
FROM(ok(z0)) → c9(FROM(z0)) 164.91/58.90
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.91/58.90
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.91/58.90
S(mark(z0)) → c12(S(z0)) 164.91/58.90
S(ok(z0)) → c13(S(z0)) 164.91/58.90
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.91/58.90
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.91/58.90
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1)) 164.91/58.90
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 164.91/58.90
ACTIVE(from(z0)) → c(S(z0)) 164.91/58.90
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.90
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.90
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.90
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 164.91/58.90
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.90
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.90
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 164.91/58.90
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 164.91/58.90
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.90
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.90
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.90
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 164.91/58.90
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.90
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 164.91/58.90
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.90
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 164.91/58.90
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.90
PROPER(from(0)) → c17(FROM(ok(0))) 164.91/58.90
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 164.91/58.90
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 164.91/58.90
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 164.91/58.90
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 164.91/58.90
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 164.91/58.90
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 164.91/58.90
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 164.91/58.90
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 164.91/58.90
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0)) 164.91/58.90
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1)) 164.91/58.90
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0))) 164.91/58.90
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.90
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0))) 164.91/58.90
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.90
PROPER(s(0)) → c19(S(ok(0))) 164.91/58.90
PROPER(sel(x0, from(z0))) → c20(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 164.91/58.90
PROPER(sel(x0, cons(z0, z1))) → c20(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 164.91/58.90
PROPER(sel(x0, s(z0))) → c20(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 164.91/58.90
PROPER(sel(x0, sel(z0, z1))) → c20(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 164.91/58.90
PROPER(sel(from(z0), x1)) → c20(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 164.91/58.90
PROPER(sel(cons(z0, z1), x1)) → c20(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 164.91/58.90
PROPER(sel(s(z0), x1)) → c20(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 164.91/58.90
PROPER(sel(sel(z0, z1), x1)) → c20(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 164.91/58.90
PROPER(sel(x0, 0)) → c20(SEL(proper(x0), ok(0)), PROPER(x0)) 164.91/58.90
PROPER(sel(0, x1)) → c20(SEL(ok(0), proper(x1)), PROPER(x1)) 164.91/58.90
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0))) 164.91/58.90
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.90
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0))) 164.91/58.90
TOP(mark(sel(z0, z1))) → c22(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.90
TOP(mark(0)) → c22(TOP(ok(0)), PROPER(0))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2)) 164.91/58.90
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0)) 164.91/58.90
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1)) 164.91/58.90
FROM(mark(z0)) → c8(FROM(z0)) 164.91/58.90
FROM(ok(z0)) → c9(FROM(z0)) 164.91/58.90
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.91/58.90
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.91/58.90
S(mark(z0)) → c12(S(z0)) 164.91/58.90
S(ok(z0)) → c13(S(z0)) 164.91/58.90
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.91/58.90
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.91/58.90
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1)) 164.91/58.90
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 164.91/58.90
ACTIVE(from(z0)) → c(S(z0)) 164.91/58.90
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.90
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.90
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.90
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 164.91/58.90
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.90
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.90
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 164.91/58.90
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 164.91/58.90
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.90
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.90
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.90
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 164.91/58.90
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.90
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 164.91/58.90
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.90
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 164.91/58.90
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.90
PROPER(from(0)) → c17(FROM(ok(0))) 164.91/58.90
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 164.91/58.90
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 164.91/58.90
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 164.91/58.90
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 164.91/58.90
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 164.91/58.90
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 164.91/58.90
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 164.91/58.90
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 164.91/58.90
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0)) 164.91/58.90
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1)) 164.91/58.90
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0))) 164.91/58.90
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.90
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0))) 164.91/58.90
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.90
PROPER(s(0)) → c19(S(ok(0))) 164.91/58.90
PROPER(sel(x0, from(z0))) → c20(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 164.91/58.90
PROPER(sel(x0, cons(z0, z1))) → c20(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 164.91/58.90
PROPER(sel(x0, s(z0))) → c20(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 164.91/58.90
PROPER(sel(x0, sel(z0, z1))) → c20(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 164.91/58.90
PROPER(sel(from(z0), x1)) → c20(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 164.91/58.90
PROPER(sel(cons(z0, z1), x1)) → c20(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 164.91/58.90
PROPER(sel(s(z0), x1)) → c20(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 164.91/58.90
PROPER(sel(sel(z0, z1), x1)) → c20(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 164.91/58.90
PROPER(sel(x0, 0)) → c20(SEL(proper(x0), ok(0)), PROPER(x0)) 164.91/58.90
PROPER(sel(0, x1)) → c20(SEL(ok(0), proper(x1)), PROPER(x1)) 164.91/58.90
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0))) 164.91/58.90
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.90
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0))) 164.91/58.90
TOP(mark(sel(z0, z1))) → c22(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.90
TOP(mark(0)) → c22(TOP(ok(0)), PROPER(0))

K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, TOP, PROPER

Compound Symbols:

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

164.91/58.90

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

Removed 1 trailing tuple parts

164.91/58.90

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 164.91/58.90
active(sel(0, cons(z0, z1))) → mark(z0) 164.91/58.90
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 164.91/58.90
active(from(z0)) → from(active(z0)) 164.91/58.90
active(cons(z0, z1)) → cons(active(z0), z1) 164.91/58.90
active(s(z0)) → s(active(z0)) 164.91/58.90
active(sel(z0, z1)) → sel(active(z0), z1) 164.91/58.90
active(sel(z0, z1)) → sel(z0, active(z1)) 164.91/58.90
from(mark(z0)) → mark(from(z0)) 164.91/58.90
from(ok(z0)) → ok(from(z0)) 164.91/58.90
cons(mark(z0), z1) → mark(cons(z0, z1)) 164.91/58.90
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 164.91/58.90
s(mark(z0)) → mark(s(z0)) 164.91/58.90
s(ok(z0)) → ok(s(z0)) 164.91/58.90
sel(mark(z0), z1) → mark(sel(z0, z1)) 164.91/58.90
sel(z0, mark(z1)) → mark(sel(z0, z1)) 164.91/58.90
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 164.91/58.90
proper(from(z0)) → from(proper(z0)) 164.91/58.90
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 164.91/58.90
proper(s(z0)) → s(proper(z0)) 164.91/58.90
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 164.91/58.90
proper(0) → ok(0) 164.91/58.90
top(mark(z0)) → top(proper(z0)) 164.91/58.90
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2)) 164.91/58.90
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0)) 164.91/58.90
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1)) 164.91/58.90
FROM(mark(z0)) → c8(FROM(z0)) 164.91/58.90
FROM(ok(z0)) → c9(FROM(z0)) 164.91/58.90
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.91/58.90
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.91/58.90
S(mark(z0)) → c12(S(z0)) 164.91/58.90
S(ok(z0)) → c13(S(z0)) 164.91/58.90
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.91/58.90
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.91/58.90
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1)) 164.91/58.90
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 164.91/58.90
ACTIVE(from(z0)) → c(S(z0)) 164.91/58.90
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.90
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.90
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.90
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 164.91/58.90
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.90
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.90
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 164.91/58.90
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 164.91/58.90
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.90
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.90
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.90
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 164.91/58.90
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.90
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 164.91/58.90
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.90
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 164.91/58.90
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.90
PROPER(from(0)) → c17(FROM(ok(0))) 164.91/58.90
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 164.91/58.90
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 164.91/58.90
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 164.91/58.90
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 164.91/58.90
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 164.91/58.90
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 164.91/58.90
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 164.91/58.90
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 164.91/58.90
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0)) 164.91/58.90
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1)) 164.91/58.90
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0))) 164.91/58.90
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.90
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0))) 164.91/58.90
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.90
PROPER(s(0)) → c19(S(ok(0))) 164.91/58.90
PROPER(sel(x0, from(z0))) → c20(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 164.91/58.90
PROPER(sel(x0, cons(z0, z1))) → c20(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 164.91/58.90
PROPER(sel(x0, s(z0))) → c20(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 164.91/58.90
PROPER(sel(x0, sel(z0, z1))) → c20(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 164.91/58.90
PROPER(sel(from(z0), x1)) → c20(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 164.91/58.90
PROPER(sel(cons(z0, z1), x1)) → c20(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 164.91/58.90
PROPER(sel(s(z0), x1)) → c20(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 164.91/58.90
PROPER(sel(sel(z0, z1), x1)) → c20(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 164.91/58.90
PROPER(sel(x0, 0)) → c20(SEL(proper(x0), ok(0)), PROPER(x0)) 164.91/58.90
PROPER(sel(0, x1)) → c20(SEL(ok(0), proper(x1)), PROPER(x1)) 164.91/58.90
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0))) 164.91/58.90
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.90
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0))) 164.91/58.90
TOP(mark(sel(z0, z1))) → c22(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.90
TOP(mark(0)) → c22(TOP(ok(0)))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2)) 164.91/58.90
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0)) 164.91/58.90
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1)) 164.91/58.90
FROM(mark(z0)) → c8(FROM(z0)) 164.91/58.90
FROM(ok(z0)) → c9(FROM(z0)) 164.91/58.90
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.91/58.90
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.91/58.90
S(mark(z0)) → c12(S(z0)) 164.91/58.90
S(ok(z0)) → c13(S(z0)) 164.91/58.90
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.91/58.90
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.91/58.90
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1)) 164.91/58.90
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 164.91/58.90
ACTIVE(from(z0)) → c(S(z0)) 164.91/58.90
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.90
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.90
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.90
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 164.91/58.90
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.90
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.90
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 164.91/58.90
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 164.91/58.90
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.90
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.90
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.90
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 164.91/58.90
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.90
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 164.91/58.90
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.90
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 164.91/58.90
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.90
PROPER(from(0)) → c17(FROM(ok(0))) 164.91/58.90
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 164.91/58.90
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 164.91/58.90
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 164.91/58.90
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 164.91/58.90
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 164.91/58.90
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 164.91/58.90
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 164.91/58.90
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 164.91/58.90
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0)) 164.91/58.90
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1)) 164.91/58.90
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0))) 164.91/58.90
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.90
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0))) 164.91/58.90
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.90
PROPER(s(0)) → c19(S(ok(0))) 164.91/58.90
PROPER(sel(x0, from(z0))) → c20(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 164.91/58.90
PROPER(sel(x0, cons(z0, z1))) → c20(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 164.91/58.90
PROPER(sel(x0, s(z0))) → c20(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 164.91/58.90
PROPER(sel(x0, sel(z0, z1))) → c20(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 164.91/58.90
PROPER(sel(from(z0), x1)) → c20(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 164.91/58.90
PROPER(sel(cons(z0, z1), x1)) → c20(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 164.91/58.90
PROPER(sel(s(z0), x1)) → c20(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 164.91/58.90
PROPER(sel(sel(z0, z1), x1)) → c20(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 164.91/58.90
PROPER(sel(x0, 0)) → c20(SEL(proper(x0), ok(0)), PROPER(x0)) 164.91/58.90
PROPER(sel(0, x1)) → c20(SEL(ok(0), proper(x1)), PROPER(x1)) 164.91/58.90
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0))) 164.91/58.90
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.90
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0))) 164.91/58.90
TOP(mark(sel(z0, z1))) → c22(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.90
TOP(mark(0)) → c22(TOP(ok(0)))

K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, TOP, PROPER

Compound Symbols:

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

164.91/58.90

(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.

TOP(mark(0)) → c22(TOP(ok(0)))

We considered the (Usable) Rules:

proper(from(z0)) → from(proper(z0)) 164.91/58.90
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 164.91/58.90
proper(s(z0)) → s(proper(z0)) 164.91/58.90
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 164.91/58.90
proper(0) → ok(0) 164.91/58.90
sel(mark(z0), z1) → mark(sel(z0, z1)) 164.91/58.90
sel(z0, mark(z1)) → mark(sel(z0, z1)) 164.91/58.90
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 164.91/58.90
s(mark(z0)) → mark(s(z0)) 164.91/58.90
s(ok(z0)) → ok(s(z0)) 164.91/58.90
cons(mark(z0), z1) → mark(cons(z0, z1)) 164.91/58.90
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 164.91/58.90
from(mark(z0)) → mark(from(z0)) 164.91/58.90
from(ok(z0)) → ok(from(z0)) 164.91/58.90
active(from(z0)) → mark(cons(z0, from(s(z0)))) 164.91/58.90
active(sel(0, cons(z0, z1))) → mark(z0) 164.91/58.90
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 164.91/58.90
active(from(z0)) → from(active(z0)) 164.91/58.90
active(cons(z0, z1)) → cons(active(z0), z1) 164.91/58.90
active(s(z0)) → s(active(z0)) 164.91/58.90
active(sel(z0, z1)) → sel(active(z0), z1) 164.91/58.90
active(sel(z0, z1)) → sel(z0, active(z1))

And the Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2)) 164.91/58.90
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0)) 164.91/58.90
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1)) 164.91/58.90
FROM(mark(z0)) → c8(FROM(z0)) 164.91/58.90
FROM(ok(z0)) → c9(FROM(z0)) 164.91/58.90
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.91/58.90
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.91/58.90
S(mark(z0)) → c12(S(z0)) 164.91/58.90
S(ok(z0)) → c13(S(z0)) 164.91/58.90
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.91/58.90
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.91/58.90
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1)) 164.91/58.90
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 164.91/58.90
ACTIVE(from(z0)) → c(S(z0)) 164.91/58.90
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.90
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.90
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.90
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 164.91/58.90
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.90
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.90
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 164.91/58.90
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 164.91/58.90
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.90
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.90
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.90
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 164.91/58.90
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.90
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 164.91/58.90
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.90
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 164.91/58.90
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.90
PROPER(from(0)) → c17(FROM(ok(0))) 164.91/58.90
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 164.91/58.90
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 164.91/58.90
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 164.91/58.90
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 164.91/58.90
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 164.91/58.90
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 164.91/58.90
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 164.91/58.90
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 164.91/58.90
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0)) 164.91/58.90
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1)) 164.91/58.90
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0))) 164.91/58.90
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.90
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0))) 164.91/58.90
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.90
PROPER(s(0)) → c19(S(ok(0))) 164.91/58.90
PROPER(sel(x0, from(z0))) → c20(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 164.91/58.90
PROPER(sel(x0, cons(z0, z1))) → c20(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 164.91/58.90
PROPER(sel(x0, s(z0))) → c20(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 164.91/58.90
PROPER(sel(x0, sel(z0, z1))) → c20(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 164.91/58.90
PROPER(sel(from(z0), x1)) → c20(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 164.91/58.90
PROPER(sel(cons(z0, z1), x1)) → c20(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 164.91/58.90
PROPER(sel(s(z0), x1)) → c20(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 164.91/58.90
PROPER(sel(sel(z0, z1), x1)) → c20(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 164.91/58.90
PROPER(sel(x0, 0)) → c20(SEL(proper(x0), ok(0)), PROPER(x0)) 164.91/58.90
PROPER(sel(0, x1)) → c20(SEL(ok(0), proper(x1)), PROPER(x1)) 164.91/58.90
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0))) 164.91/58.90
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.90
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0))) 164.91/58.90
TOP(mark(sel(z0, z1))) → c22(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.90
TOP(mark(0)) → c22(TOP(ok(0)))

The order we found is given by the following interpretation:
Polynomial interpretation : 164.91/58.90

POL(0) = 0 164.91/58.90
POL(ACTIVE(x₁)) = 0 164.91/58.90
POL(CONS(x₁, x₂)) = 0 164.91/58.90
POL(FROM(x₁)) = 0 164.91/58.90
POL(PROPER(x₁)) = 0 164.91/58.90
POL(S(x₁)) = 0 164.91/58.90
POL(SEL(x₁, x₂)) = 0 164.91/58.90
POL(TOP(x₁)) = [4]x₁ 164.91/58.90
POL(active(x₁)) = x₁ 164.91/58.90
POL(c(x₁)) = x₁ 164.91/58.90
POL(c10(x₁)) = x₁ 164.91/58.90
POL(c11(x₁)) = x₁ 164.91/58.90
POL(c12(x₁)) = x₁ 164.91/58.90
POL(c13(x₁)) = x₁ 164.91/58.90
POL(c14(x₁)) = x₁ 164.91/58.90
POL(c15(x₁)) = x₁ 164.91/58.90
POL(c16(x₁)) = x₁ 164.91/58.90
POL(c17(x₁)) = x₁ 164.91/58.90
POL(c17(x₁, x₂)) = x₁ + x₂ 164.91/58.90
POL(c18(x₁, x₂)) = x₁ + x₂ 164.91/58.90
POL(c18(x₁, x₂, x₃)) = x₁ + x₂ + x₃ 164.91/58.90
POL(c19(x₁)) = x₁ 164.91/58.90
POL(c19(x₁, x₂)) = x₁ + x₂ 164.91/58.90
POL(c2(x₁)) = x₁ 164.91/58.90
POL(c20(x₁, x₂)) = x₁ + x₂ 164.91/58.90
POL(c20(x₁, x₂, x₃)) = x₁ + x₂ + x₃ 164.91/58.90
POL(c22(x₁)) = x₁ 164.91/58.90
POL(c22(x₁, x₂)) = x₁ + x₂ 164.91/58.90
POL(c23(x₁, x₂)) = x₁ + x₂ 164.91/58.90
POL(c3(x₁, x₂)) = x₁ + x₂ 164.91/58.90
POL(c4(x₁, x₂)) = x₁ + x₂ 164.91/58.90
POL(c5(x₁, x₂)) = x₁ + x₂ 164.91/58.90
POL(c6(x₁, x₂)) = x₁ + x₂ 164.91/58.90
POL(c7(x₁, x₂)) = x₁ + x₂ 164.91/58.90
POL(c8(x₁)) = x₁ 164.91/58.90
POL(c9(x₁)) = x₁ 164.91/58.90
POL(cons(x₁, x₂)) = [2] 164.91/58.90
POL(from(x₁)) = [2] 164.91/58.90
POL(mark(x₁)) = [2] 164.91/58.90
POL(ok(x₁)) = x₁ 164.91/58.90
POL(proper(x₁)) = 0 164.91/58.90
POL(s(x₁)) = [2] 164.91/58.90
POL(sel(x₁, x₂)) = [2]

164.91/58.90

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 164.91/58.90
active(sel(0, cons(z0, z1))) → mark(z0) 164.91/58.90
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 164.91/58.90
active(from(z0)) → from(active(z0)) 164.91/58.90
active(cons(z0, z1)) → cons(active(z0), z1) 164.91/58.90
active(s(z0)) → s(active(z0)) 164.91/58.90
active(sel(z0, z1)) → sel(active(z0), z1) 164.91/58.90
active(sel(z0, z1)) → sel(z0, active(z1)) 164.91/58.90
from(mark(z0)) → mark(from(z0)) 164.91/58.90
from(ok(z0)) → ok(from(z0)) 164.91/58.90
cons(mark(z0), z1) → mark(cons(z0, z1)) 164.91/58.90
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 164.91/58.90
s(mark(z0)) → mark(s(z0)) 164.91/58.90
s(ok(z0)) → ok(s(z0)) 164.91/58.90
sel(mark(z0), z1) → mark(sel(z0, z1)) 164.91/58.90
sel(z0, mark(z1)) → mark(sel(z0, z1)) 164.91/58.90
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 164.91/58.90
proper(from(z0)) → from(proper(z0)) 164.91/58.90
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 164.91/58.90
proper(s(z0)) → s(proper(z0)) 164.91/58.90
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 164.91/58.90
proper(0) → ok(0) 164.91/58.90
top(mark(z0)) → top(proper(z0)) 164.91/58.90
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2)) 164.91/58.90
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0)) 164.91/58.90
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1)) 164.91/58.90
FROM(mark(z0)) → c8(FROM(z0)) 164.91/58.90
FROM(ok(z0)) → c9(FROM(z0)) 164.91/58.90
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.91/58.90
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.91/58.90
S(mark(z0)) → c12(S(z0)) 164.91/58.90
S(ok(z0)) → c13(S(z0)) 164.91/58.90
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.91/58.90
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.91/58.90
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1)) 164.91/58.90
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 164.91/58.90
ACTIVE(from(z0)) → c(S(z0)) 164.91/58.90
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.90
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.90
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.90
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 164.91/58.90
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.90
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.90
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 164.91/58.90
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 164.91/58.90
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.90
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.90
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.90
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 164.91/58.90
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.90
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 164.91/58.90
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.90
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 164.91/58.90
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.90
PROPER(from(0)) → c17(FROM(ok(0))) 164.91/58.90
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 164.91/58.90
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 164.91/58.90
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 164.91/58.90
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 164.91/58.90
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 164.91/58.90
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 164.91/58.90
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 164.91/58.90
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 164.91/58.90
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0)) 164.91/58.90
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1)) 164.91/58.90
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0))) 164.91/58.90
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.90
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0))) 164.91/58.90
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.90
PROPER(s(0)) → c19(S(ok(0))) 164.91/58.90
PROPER(sel(x0, from(z0))) → c20(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 164.91/58.90
PROPER(sel(x0, cons(z0, z1))) → c20(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 164.91/58.90
PROPER(sel(x0, s(z0))) → c20(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 164.91/58.90
PROPER(sel(x0, sel(z0, z1))) → c20(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 164.91/58.90
PROPER(sel(from(z0), x1)) → c20(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 164.91/58.90
PROPER(sel(cons(z0, z1), x1)) → c20(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 164.91/58.90
PROPER(sel(s(z0), x1)) → c20(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 164.91/58.90
PROPER(sel(sel(z0, z1), x1)) → c20(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 164.91/58.90
PROPER(sel(x0, 0)) → c20(SEL(proper(x0), ok(0)), PROPER(x0)) 164.91/58.90
PROPER(sel(0, x1)) → c20(SEL(ok(0), proper(x1)), PROPER(x1)) 164.91/58.90
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0))) 164.91/58.90
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.90
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0))) 164.91/58.90
TOP(mark(sel(z0, z1))) → c22(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.90
TOP(mark(0)) → c22(TOP(ok(0)))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2)) 164.91/58.90
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0)) 164.91/58.90
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1)) 164.91/58.90
FROM(mark(z0)) → c8(FROM(z0)) 164.91/58.90
FROM(ok(z0)) → c9(FROM(z0)) 164.91/58.90
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.91/58.90
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.91/58.90
S(mark(z0)) → c12(S(z0)) 164.91/58.90
S(ok(z0)) → c13(S(z0)) 164.91/58.90
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.91/58.90
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.91/58.90
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1)) 164.91/58.90
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 164.91/58.90
ACTIVE(from(z0)) → c(S(z0)) 164.91/58.90
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.90
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.90
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.90
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 164.91/58.90
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.90
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.90
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 164.91/58.90
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 164.91/58.90
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.90
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.90
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.90
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 164.91/58.90
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.90
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 164.91/58.90
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.90
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 164.91/58.90
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.90
PROPER(from(0)) → c17(FROM(ok(0))) 164.91/58.90
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 164.91/58.90
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 164.91/58.90
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 164.91/58.90
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 164.91/58.90
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 164.91/58.90
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 164.91/58.90
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 164.91/58.90
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 164.91/58.90
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0)) 164.91/58.90
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1)) 164.91/58.90
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0))) 164.91/58.90
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.90
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0))) 164.91/58.90
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.90
PROPER(s(0)) → c19(S(ok(0))) 164.91/58.90
PROPER(sel(x0, from(z0))) → c20(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 164.91/58.90
PROPER(sel(x0, cons(z0, z1))) → c20(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 164.91/58.90
PROPER(sel(x0, s(z0))) → c20(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 164.91/58.90
PROPER(sel(x0, sel(z0, z1))) → c20(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 164.91/58.90
PROPER(sel(from(z0), x1)) → c20(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 164.91/58.90
PROPER(sel(cons(z0, z1), x1)) → c20(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 164.91/58.90
PROPER(sel(s(z0), x1)) → c20(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 164.91/58.90
PROPER(sel(sel(z0, z1), x1)) → c20(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 164.91/58.90
PROPER(sel(x0, 0)) → c20(SEL(proper(x0), ok(0)), PROPER(x0)) 164.91/58.90
PROPER(sel(0, x1)) → c20(SEL(ok(0), proper(x1)), PROPER(x1)) 164.91/58.90
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0))) 164.91/58.90
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.90
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0))) 164.91/58.90
TOP(mark(sel(z0, z1))) → c22(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))

K tuples:

TOP(mark(0)) → c22(TOP(ok(0)))

Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, TOP, PROPER

Compound Symbols:

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

164.91/58.90

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

Use narrowing to replace TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(from(z0))) → c23(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.90
TOP(ok(sel(0, cons(z0, z1)))) → c23(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.90
TOP(ok(sel(s(z0), cons(z1, z2)))) → c23(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.90
TOP(ok(from(z0))) → c23(TOP(from(active(z0))), ACTIVE(from(z0))) 164.91/58.90
TOP(ok(cons(z0, z1))) → c23(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.90
TOP(ok(s(z0))) → c23(TOP(s(active(z0))), ACTIVE(s(z0))) 164.91/58.90
TOP(ok(sel(z0, z1))) → c23(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.90
TOP(ok(sel(z0, z1))) → c23(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))

164.91/58.90

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 164.91/58.90
active(sel(0, cons(z0, z1))) → mark(z0) 164.91/58.90
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 164.91/58.90
active(from(z0)) → from(active(z0)) 164.91/58.90
active(cons(z0, z1)) → cons(active(z0), z1) 164.91/58.90
active(s(z0)) → s(active(z0)) 164.91/58.90
active(sel(z0, z1)) → sel(active(z0), z1) 164.91/58.90
active(sel(z0, z1)) → sel(z0, active(z1)) 164.91/58.90
from(mark(z0)) → mark(from(z0)) 164.91/58.90
from(ok(z0)) → ok(from(z0)) 164.91/58.90
cons(mark(z0), z1) → mark(cons(z0, z1)) 164.91/58.90
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 164.91/58.90
s(mark(z0)) → mark(s(z0)) 164.91/58.90
s(ok(z0)) → ok(s(z0)) 164.91/58.90
sel(mark(z0), z1) → mark(sel(z0, z1)) 164.91/58.90
sel(z0, mark(z1)) → mark(sel(z0, z1)) 164.91/58.90
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 164.91/58.90
proper(from(z0)) → from(proper(z0)) 164.91/58.90
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 164.91/58.90
proper(s(z0)) → s(proper(z0)) 164.91/58.90
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 164.91/58.90
proper(0) → ok(0) 164.91/58.90
top(mark(z0)) → top(proper(z0)) 164.91/58.90
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2)) 164.91/58.90
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0)) 164.91/58.90
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1)) 164.91/58.90
FROM(mark(z0)) → c8(FROM(z0)) 164.91/58.90
FROM(ok(z0)) → c9(FROM(z0)) 164.91/58.90
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.91/58.90
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.91/58.90
S(mark(z0)) → c12(S(z0)) 164.91/58.90
S(ok(z0)) → c13(S(z0)) 164.91/58.90
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.91/58.90
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.91/58.90
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1)) 164.91/58.90
ACTIVE(from(z0)) → c(S(z0)) 164.91/58.90
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.90
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.90
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.90
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 164.91/58.90
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.90
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.90
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 164.91/58.90
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 164.91/58.90
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.90
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.90
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.90
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 164.91/58.90
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.90
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 164.91/58.90
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.90
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 164.91/58.90
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.90
PROPER(from(0)) → c17(FROM(ok(0))) 164.91/58.90
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 164.91/58.90
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 164.91/58.90
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 164.91/58.90
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 164.91/58.90
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 164.91/58.90
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 164.91/58.90
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 164.91/58.90
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 164.91/58.90
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0)) 164.91/58.90
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1)) 164.91/58.90
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0))) 164.91/58.90
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.90
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0))) 164.91/58.90
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.90
PROPER(s(0)) → c19(S(ok(0))) 164.91/58.90
PROPER(sel(x0, from(z0))) → c20(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 164.91/58.90
PROPER(sel(x0, cons(z0, z1))) → c20(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 164.91/58.90
PROPER(sel(x0, s(z0))) → c20(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 164.91/58.90
PROPER(sel(x0, sel(z0, z1))) → c20(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 164.91/58.90
PROPER(sel(from(z0), x1)) → c20(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 164.91/58.90
PROPER(sel(cons(z0, z1), x1)) → c20(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 164.91/58.90
PROPER(sel(s(z0), x1)) → c20(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 164.91/58.90
PROPER(sel(sel(z0, z1), x1)) → c20(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 164.91/58.90
PROPER(sel(x0, 0)) → c20(SEL(proper(x0), ok(0)), PROPER(x0)) 164.91/58.90
PROPER(sel(0, x1)) → c20(SEL(ok(0), proper(x1)), PROPER(x1)) 164.91/58.90
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0))) 164.91/58.90
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.90
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0))) 164.91/58.90
TOP(mark(sel(z0, z1))) → c22(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.90
TOP(mark(0)) → c22(TOP(ok(0))) 164.91/58.90
TOP(ok(from(z0))) → c23(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.90
TOP(ok(sel(0, cons(z0, z1)))) → c23(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.90
TOP(ok(sel(s(z0), cons(z1, z2)))) → c23(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.90
TOP(ok(from(z0))) → c23(TOP(from(active(z0))), ACTIVE(from(z0))) 164.91/58.90
TOP(ok(cons(z0, z1))) → c23(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.90
TOP(ok(s(z0))) → c23(TOP(s(active(z0))), ACTIVE(s(z0))) 164.91/58.90
TOP(ok(sel(z0, z1))) → c23(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.90
TOP(ok(sel(z0, z1))) → c23(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2)) 164.91/58.90
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0)) 164.91/58.90
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1)) 164.91/58.90
FROM(mark(z0)) → c8(FROM(z0)) 164.91/58.90
FROM(ok(z0)) → c9(FROM(z0)) 164.91/58.90
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.91/58.90
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.91/58.90
S(mark(z0)) → c12(S(z0)) 164.91/58.90
S(ok(z0)) → c13(S(z0)) 164.91/58.90
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.91/58.90
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.91/58.90
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1)) 164.91/58.90
ACTIVE(from(z0)) → c(S(z0)) 164.91/58.90
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.90
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.90
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.90
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 164.91/58.90
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.90
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.90
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 164.91/58.90
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 164.91/58.90
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.90
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.90
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.90
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 164.91/58.90
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.90
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.90
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 164.91/58.90
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.90
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 164.91/58.90
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.90
PROPER(from(0)) → c17(FROM(ok(0))) 164.91/58.90
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 164.91/58.90
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 164.91/58.90
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 164.91/58.90
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 164.91/58.90
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 164.91/58.90
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 164.91/58.90
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 164.91/58.90
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 164.91/58.90
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0)) 164.91/58.90
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1)) 164.91/58.90
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0))) 164.91/58.90
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.90
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0))) 164.91/58.90
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.90
PROPER(s(0)) → c19(S(ok(0))) 164.91/58.90
PROPER(sel(x0, from(z0))) → c20(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 164.91/58.90
PROPER(sel(x0, cons(z0, z1))) → c20(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 164.91/58.90
PROPER(sel(x0, s(z0))) → c20(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 164.91/58.90
PROPER(sel(x0, sel(z0, z1))) → c20(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 164.91/58.90
PROPER(sel(from(z0), x1)) → c20(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 164.91/58.90
PROPER(sel(cons(z0, z1), x1)) → c20(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 164.91/58.90
PROPER(sel(s(z0), x1)) → c20(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 164.91/58.90
PROPER(sel(sel(z0, z1), x1)) → c20(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 164.91/58.90
PROPER(sel(x0, 0)) → c20(SEL(proper(x0), ok(0)), PROPER(x0)) 164.91/58.90
PROPER(sel(0, x1)) → c20(SEL(ok(0), proper(x1)), PROPER(x1)) 164.91/58.90
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0))) 164.91/58.90
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.90
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0))) 164.91/58.90
TOP(mark(sel(z0, z1))) → c22(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.90
TOP(ok(from(z0))) → c23(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.90
TOP(ok(sel(0, cons(z0, z1)))) → c23(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.90
TOP(ok(sel(s(z0), cons(z1, z2)))) → c23(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.90
TOP(ok(from(z0))) → c23(TOP(from(active(z0))), ACTIVE(from(z0))) 164.91/58.90
TOP(ok(cons(z0, z1))) → c23(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.90
TOP(ok(s(z0))) → c23(TOP(s(active(z0))), ACTIVE(s(z0))) 164.91/58.90
TOP(ok(sel(z0, z1))) → c23(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.90
TOP(ok(sel(z0, z1))) → c23(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))

K tuples:

TOP(mark(0)) → c22(TOP(ok(0)))

Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, SEL, PROPER, TOP

Compound Symbols:

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

164.91/58.90

(35) CdtUnreachableProof (EQUIVALENT transformation)

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

ACTIVE(sel(s(z0), cons(z1, z2))) → c2(SEL(z0, z2)) 164.91/58.90
ACTIVE(sel(z0, z1)) → c6(SEL(active(z0), z1), ACTIVE(z0)) 164.91/58.90
ACTIVE(sel(z0, z1)) → c7(SEL(z0, active(z1)), ACTIVE(z1)) 164.91/58.90
ACTIVE(from(z0)) → c(S(z0)) 164.91/58.90
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(from(sel(0, cons(z0, z1)))) → c3(FROM(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.90
ACTIVE(from(sel(s(z0), cons(z1, z2)))) → c3(FROM(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.90
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 164.91/58.90
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.90
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 164.91/58.91
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.91
ACTIVE(from(sel(z0, z1))) → c3(FROM(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.91
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 164.91/58.91
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c4(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.91
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.91
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 164.91/58.91
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 164.91/58.91
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 164.91/58.91
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 164.91/58.91
ACTIVE(cons(sel(z0, z1), x1)) → c4(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 164.91/58.91
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.91
ACTIVE(s(sel(0, cons(z0, z1)))) → c5(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.91
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c5(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.91
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 164.91/58.91
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.91
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 164.91/58.91
ACTIVE(s(sel(z0, z1))) → c5(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.91
ACTIVE(s(sel(z0, z1))) → c5(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 164.91/58.91
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 164.91/58.91
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.91
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 164.91/58.91
PROPER(from(sel(z0, z1))) → c17(FROM(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.91
PROPER(from(0)) → c17(FROM(ok(0))) 164.91/58.91
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 164.91/58.91
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 164.91/58.91
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 164.91/58.91
PROPER(cons(x0, sel(z0, z1))) → c18(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 164.91/58.91
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 164.91/58.91
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 164.91/58.91
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 164.91/58.91
PROPER(cons(sel(z0, z1), x1)) → c18(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 164.91/58.91
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0)) 164.91/58.91
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1)) 164.91/58.91
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0))) 164.91/58.91
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.91
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0))) 164.91/58.91
PROPER(s(sel(z0, z1))) → c19(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.91
PROPER(s(0)) → c19(S(ok(0))) 164.91/58.91
PROPER(sel(x0, from(z0))) → c20(SEL(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 164.91/58.91
PROPER(sel(x0, cons(z0, z1))) → c20(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 164.91/58.91
PROPER(sel(x0, s(z0))) → c20(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 164.91/58.91
PROPER(sel(x0, sel(z0, z1))) → c20(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 164.91/58.91
PROPER(sel(from(z0), x1)) → c20(SEL(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 164.91/58.91
PROPER(sel(cons(z0, z1), x1)) → c20(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 164.91/58.91
PROPER(sel(s(z0), x1)) → c20(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 164.91/58.91
PROPER(sel(sel(z0, z1), x1)) → c20(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 164.91/58.91
PROPER(sel(x0, 0)) → c20(SEL(proper(x0), ok(0)), PROPER(x0)) 164.91/58.91
PROPER(sel(0, x1)) → c20(SEL(ok(0), proper(x1)), PROPER(x1)) 164.91/58.91
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0))) 164.91/58.91
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 164.91/58.91
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0))) 164.91/58.91
TOP(mark(sel(z0, z1))) → c22(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 164.91/58.91
TOP(ok(from(z0))) → c23(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 164.91/58.91
TOP(ok(sel(0, cons(z0, z1)))) → c23(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 164.91/58.91
TOP(ok(sel(s(z0), cons(z1, z2)))) → c23(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 164.91/58.91
TOP(ok(from(z0))) → c23(TOP(from(active(z0))), ACTIVE(from(z0))) 164.91/58.91
TOP(ok(cons(z0, z1))) → c23(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 164.91/58.91
TOP(ok(s(z0))) → c23(TOP(s(active(z0))), ACTIVE(s(z0))) 164.91/58.91
TOP(ok(sel(z0, z1))) → c23(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 164.91/58.91
TOP(ok(sel(z0, z1))) → c23(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))

164.91/58.91

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 164.91/58.91
active(sel(0, cons(z0, z1))) → mark(z0) 164.91/58.91
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 164.91/58.91
active(from(z0)) → from(active(z0)) 164.91/58.91
active(cons(z0, z1)) → cons(active(z0), z1) 164.91/58.91
active(s(z0)) → s(active(z0)) 164.91/58.91
active(sel(z0, z1)) → sel(active(z0), z1) 164.91/58.91
active(sel(z0, z1)) → sel(z0, active(z1)) 164.91/58.91
from(mark(z0)) → mark(from(z0)) 164.91/58.91
from(ok(z0)) → ok(from(z0)) 164.91/58.91
cons(mark(z0), z1) → mark(cons(z0, z1)) 164.91/58.91
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 164.91/58.91
s(mark(z0)) → mark(s(z0)) 164.91/58.91
s(ok(z0)) → ok(s(z0)) 164.91/58.91
sel(mark(z0), z1) → mark(sel(z0, z1)) 164.91/58.91
sel(z0, mark(z1)) → mark(sel(z0, z1)) 164.91/58.91
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 164.91/58.91
proper(from(z0)) → from(proper(z0)) 164.91/58.91
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 164.91/58.91
proper(s(z0)) → s(proper(z0)) 164.91/58.91
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 164.91/58.91
proper(0) → ok(0) 164.91/58.91
top(mark(z0)) → top(proper(z0)) 164.91/58.91
top(ok(z0)) → top(active(z0))

Tuples:

FROM(mark(z0)) → c8(FROM(z0)) 164.91/58.91
FROM(ok(z0)) → c9(FROM(z0)) 164.91/58.91
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.91/58.91
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.91/58.91
S(mark(z0)) → c12(S(z0)) 164.91/58.91
S(ok(z0)) → c13(S(z0)) 164.91/58.91
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.91/58.91
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.91/58.91
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1)) 164.91/58.91
TOP(mark(0)) → c22(TOP(ok(0)))

S tuples:

FROM(mark(z0)) → c8(FROM(z0)) 164.91/58.91
FROM(ok(z0)) → c9(FROM(z0)) 164.91/58.91
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.91/58.91
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.91/58.91
S(mark(z0)) → c12(S(z0)) 164.91/58.91
S(ok(z0)) → c13(S(z0)) 164.91/58.91
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.91/58.91
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.91/58.91
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))

K tuples:

TOP(mark(0)) → c22(TOP(ok(0)))

Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

FROM, CONS, S, SEL, TOP

Compound Symbols:

c8, c9, c10, c11, c12, c13, c14, c15, c16, c22

164.91/58.91

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

Removed 1 trailing tuple parts

164.91/58.91

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 164.91/58.91
active(sel(0, cons(z0, z1))) → mark(z0) 164.91/58.91
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 164.91/58.91
active(from(z0)) → from(active(z0)) 164.91/58.91
active(cons(z0, z1)) → cons(active(z0), z1) 164.91/58.91
active(s(z0)) → s(active(z0)) 164.91/58.91
active(sel(z0, z1)) → sel(active(z0), z1) 164.91/58.91
active(sel(z0, z1)) → sel(z0, active(z1)) 164.91/58.91
from(mark(z0)) → mark(from(z0)) 164.91/58.91
from(ok(z0)) → ok(from(z0)) 164.91/58.91
cons(mark(z0), z1) → mark(cons(z0, z1)) 164.91/58.91
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 164.91/58.91
s(mark(z0)) → mark(s(z0)) 164.91/58.91
s(ok(z0)) → ok(s(z0)) 164.91/58.91
sel(mark(z0), z1) → mark(sel(z0, z1)) 164.91/58.91
sel(z0, mark(z1)) → mark(sel(z0, z1)) 164.91/58.91
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 164.91/58.91
proper(from(z0)) → from(proper(z0)) 164.91/58.91
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 164.91/58.91
proper(s(z0)) → s(proper(z0)) 164.91/58.91
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 164.91/58.91
proper(0) → ok(0) 164.91/58.91
top(mark(z0)) → top(proper(z0)) 164.91/58.91
top(ok(z0)) → top(active(z0))

Tuples:

FROM(mark(z0)) → c8(FROM(z0)) 164.91/58.91
FROM(ok(z0)) → c9(FROM(z0)) 164.91/58.91
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.91/58.91
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.91/58.91
S(mark(z0)) → c12(S(z0)) 164.91/58.91
S(ok(z0)) → c13(S(z0)) 164.91/58.91
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.91/58.91
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.91/58.91
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1)) 164.91/58.91
TOP(mark(0)) → c22

S tuples:

FROM(mark(z0)) → c8(FROM(z0)) 164.91/58.91
FROM(ok(z0)) → c9(FROM(z0)) 164.91/58.91
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.91/58.91
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.91/58.91
S(mark(z0)) → c12(S(z0)) 164.91/58.91
S(ok(z0)) → c13(S(z0)) 164.91/58.91
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.91/58.91
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.91/58.91
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))

K tuples:

TOP(mark(0)) → c22

Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

FROM, CONS, S, SEL, TOP

Compound Symbols:

c8, c9, c10, c11, c12, c13, c14, c15, c16, c22

164.91/58.91

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

Removed 1 trailing nodes:

TOP(mark(0)) → c22

164.91/58.91

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 164.91/58.91
active(sel(0, cons(z0, z1))) → mark(z0) 164.91/58.91
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 164.91/58.91
active(from(z0)) → from(active(z0)) 164.91/58.91
active(cons(z0, z1)) → cons(active(z0), z1) 164.91/58.91
active(s(z0)) → s(active(z0)) 164.91/58.91
active(sel(z0, z1)) → sel(active(z0), z1) 164.91/58.91
active(sel(z0, z1)) → sel(z0, active(z1)) 164.91/58.91
from(mark(z0)) → mark(from(z0)) 164.91/58.91
from(ok(z0)) → ok(from(z0)) 164.91/58.91
cons(mark(z0), z1) → mark(cons(z0, z1)) 164.91/58.91
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 164.91/58.91
s(mark(z0)) → mark(s(z0)) 164.91/58.91
s(ok(z0)) → ok(s(z0)) 164.91/58.91
sel(mark(z0), z1) → mark(sel(z0, z1)) 164.91/58.91
sel(z0, mark(z1)) → mark(sel(z0, z1)) 164.91/58.91
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 164.91/58.91
proper(from(z0)) → from(proper(z0)) 164.91/58.91
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 164.91/58.91
proper(s(z0)) → s(proper(z0)) 164.91/58.91
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 164.91/58.91
proper(0) → ok(0) 164.91/58.91
top(mark(z0)) → top(proper(z0)) 164.91/58.91
top(ok(z0)) → top(active(z0))

Tuples:

FROM(mark(z0)) → c8(FROM(z0)) 164.91/58.91
FROM(ok(z0)) → c9(FROM(z0)) 164.91/58.91
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.91/58.91
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.91/58.91
S(mark(z0)) → c12(S(z0)) 164.91/58.91
S(ok(z0)) → c13(S(z0)) 164.91/58.91
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.91/58.91
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.91/58.91
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))

S tuples:

FROM(mark(z0)) → c8(FROM(z0)) 164.91/58.91
FROM(ok(z0)) → c9(FROM(z0)) 164.91/58.91
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.91/58.91
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.91/58.91
S(mark(z0)) → c12(S(z0)) 164.91/58.91
S(ok(z0)) → c13(S(z0)) 164.91/58.91
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.91/58.91
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.91/58.91
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))

K tuples:none
Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

FROM, CONS, S, SEL

Compound Symbols:

c8, c9, c10, c11, c12, c13, c14, c15, c16

164.91/58.91

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

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

S(mark(z0)) → c12(S(z0)) 164.91/58.91
S(ok(z0)) → c13(S(z0)) 164.91/58.91
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.91/58.91
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.91/58.91
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))

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

FROM(mark(z0)) → c8(FROM(z0)) 164.91/58.91
FROM(ok(z0)) → c9(FROM(z0)) 164.91/58.91
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.91/58.91
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.91/58.91
S(mark(z0)) → c12(S(z0)) 164.91/58.91
S(ok(z0)) → c13(S(z0)) 164.91/58.91
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.91/58.91
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.91/58.91
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 164.91/58.91

POL(CONS(x₁, x₂)) = 0 164.91/58.91
POL(FROM(x₁)) = 0 164.91/58.91
POL(S(x₁)) = x₁ 164.91/58.91
POL(SEL(x₁, x₂)) = x₁ + x₂ 164.91/58.91
POL(c10(x₁)) = x₁ 164.91/58.91
POL(c11(x₁)) = x₁ 164.91/58.91
POL(c12(x₁)) = x₁ 164.91/58.91
POL(c13(x₁)) = x₁ 164.91/58.91
POL(c14(x₁)) = x₁ 164.91/58.91
POL(c15(x₁)) = x₁ 164.91/58.91
POL(c16(x₁)) = x₁ 164.91/58.91
POL(c8(x₁)) = x₁ 164.91/58.91
POL(c9(x₁)) = x₁ 164.91/58.91
POL(mark(x₁)) = [1] + x₁ 164.91/58.91
POL(ok(x₁)) = [1] + x₁

164.91/58.91

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 164.91/58.91
active(sel(0, cons(z0, z1))) → mark(z0) 164.91/58.91
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 164.91/58.91
active(from(z0)) → from(active(z0)) 164.91/58.91
active(cons(z0, z1)) → cons(active(z0), z1) 164.91/58.91
active(s(z0)) → s(active(z0)) 164.91/58.91
active(sel(z0, z1)) → sel(active(z0), z1) 164.91/58.91
active(sel(z0, z1)) → sel(z0, active(z1)) 164.91/58.91
from(mark(z0)) → mark(from(z0)) 164.91/58.91
from(ok(z0)) → ok(from(z0)) 164.91/58.91
cons(mark(z0), z1) → mark(cons(z0, z1)) 164.91/58.91
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 164.91/58.91
s(mark(z0)) → mark(s(z0)) 164.91/58.91
s(ok(z0)) → ok(s(z0)) 164.91/58.91
sel(mark(z0), z1) → mark(sel(z0, z1)) 164.91/58.91
sel(z0, mark(z1)) → mark(sel(z0, z1)) 164.91/58.91
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 164.91/58.91
proper(from(z0)) → from(proper(z0)) 164.91/58.91
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 164.91/58.91
proper(s(z0)) → s(proper(z0)) 164.91/58.91
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 164.91/58.91
proper(0) → ok(0) 164.91/58.91
top(mark(z0)) → top(proper(z0)) 164.91/58.91
top(ok(z0)) → top(active(z0))

Tuples:

FROM(mark(z0)) → c8(FROM(z0)) 164.91/58.91
FROM(ok(z0)) → c9(FROM(z0)) 164.91/58.91
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.91/58.91
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.91/58.91
S(mark(z0)) → c12(S(z0)) 164.91/58.91
S(ok(z0)) → c13(S(z0)) 164.91/58.91
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.91/58.91
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.91/58.91
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))

S tuples:

FROM(mark(z0)) → c8(FROM(z0)) 164.91/58.91
FROM(ok(z0)) → c9(FROM(z0)) 164.91/58.91
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.91/58.91
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))

K tuples:

S(mark(z0)) → c12(S(z0)) 164.91/58.91
S(ok(z0)) → c13(S(z0)) 164.91/58.91
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.91/58.91
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.91/58.91
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))

Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

FROM, CONS, S, SEL

Compound Symbols:

c8, c9, c10, c11, c12, c13, c14, c15, c16

164.91/58.91

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

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

FROM(mark(z0)) → c8(FROM(z0))

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

FROM(mark(z0)) → c8(FROM(z0)) 164.91/58.91
FROM(ok(z0)) → c9(FROM(z0)) 164.91/58.91
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.91/58.91
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.91/58.91
S(mark(z0)) → c12(S(z0)) 164.91/58.91
S(ok(z0)) → c13(S(z0)) 164.91/58.91
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.91/58.91
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.91/58.91
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 164.91/58.91

POL(CONS(x₁, x₂)) = x₂ 164.91/58.91
POL(FROM(x₁)) = [2]x₁ 164.91/58.91
POL(S(x₁)) = [3]x₁ 164.91/58.91
POL(SEL(x₁, x₂)) = [5]x₁ + [3]x₂ 164.91/58.91
POL(c10(x₁)) = x₁ 164.91/58.91
POL(c11(x₁)) = x₁ 164.91/58.91
POL(c12(x₁)) = x₁ 164.91/58.91
POL(c13(x₁)) = x₁ 164.91/58.91
POL(c14(x₁)) = x₁ 164.91/58.91
POL(c15(x₁)) = x₁ 164.91/58.91
POL(c16(x₁)) = x₁ 164.91/58.91
POL(c8(x₁)) = x₁ 164.91/58.91
POL(c9(x₁)) = x₁ 164.91/58.91
POL(mark(x₁)) = [1] + x₁ 164.91/58.91
POL(ok(x₁)) = x₁

164.91/58.91

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 164.91/58.91
active(sel(0, cons(z0, z1))) → mark(z0) 164.91/58.91
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 164.91/58.91
active(from(z0)) → from(active(z0)) 164.91/58.91
active(cons(z0, z1)) → cons(active(z0), z1) 164.91/58.91
active(s(z0)) → s(active(z0)) 164.91/58.91
active(sel(z0, z1)) → sel(active(z0), z1) 164.91/58.91
active(sel(z0, z1)) → sel(z0, active(z1)) 164.91/58.91
from(mark(z0)) → mark(from(z0)) 164.91/58.91
from(ok(z0)) → ok(from(z0)) 164.91/58.91
cons(mark(z0), z1) → mark(cons(z0, z1)) 164.91/58.91
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 164.91/58.91
s(mark(z0)) → mark(s(z0)) 164.91/58.91
s(ok(z0)) → ok(s(z0)) 164.91/58.91
sel(mark(z0), z1) → mark(sel(z0, z1)) 164.91/58.91
sel(z0, mark(z1)) → mark(sel(z0, z1)) 164.91/58.91
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 164.91/58.91
proper(from(z0)) → from(proper(z0)) 164.91/58.91
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 164.91/58.91
proper(s(z0)) → s(proper(z0)) 164.91/58.91
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 164.91/58.91
proper(0) → ok(0) 164.91/58.91
top(mark(z0)) → top(proper(z0)) 164.91/58.91
top(ok(z0)) → top(active(z0))

Tuples:

FROM(mark(z0)) → c8(FROM(z0)) 164.91/58.91
FROM(ok(z0)) → c9(FROM(z0)) 164.91/58.91
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.91/58.91
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.91/58.91
S(mark(z0)) → c12(S(z0)) 164.91/58.91
S(ok(z0)) → c13(S(z0)) 164.91/58.91
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.91/58.91
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.91/58.91
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))

S tuples:

FROM(ok(z0)) → c9(FROM(z0)) 164.91/58.91
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.91/58.91
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))

K tuples:

S(mark(z0)) → c12(S(z0)) 164.91/58.91
S(ok(z0)) → c13(S(z0)) 164.91/58.91
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.91/58.91
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.91/58.91
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1)) 164.91/58.91
FROM(mark(z0)) → c8(FROM(z0))

Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

FROM, CONS, S, SEL

Compound Symbols:

c8, c9, c10, c11, c12, c13, c14, c15, c16

164.91/58.91

(45) 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) → c10(CONS(z0, z1))

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

FROM(mark(z0)) → c8(FROM(z0)) 164.91/58.91
FROM(ok(z0)) → c9(FROM(z0)) 164.91/58.91
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 164.91/58.91
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 164.91/58.91
S(mark(z0)) → c12(S(z0)) 164.91/58.91
S(ok(z0)) → c13(S(z0)) 164.91/58.91
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 164.91/58.91
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 164.91/58.91
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 164.91/58.91

POL(CONS(x₁, x₂)) = x₁ + x₂ 164.91/58.91
POL(FROM(x₁)) = x₁ 164.91/58.91
POL(S(x₁)) = [3]x₁ 164.91/58.91
POL(SEL(x₁, x₂)) = [3]x₁ + [3]x₂ 164.91/58.91
POL(c10(x₁)) = x₁ 164.91/58.91
POL(c11(x₁)) = x₁ 164.91/58.91
POL(c12(x₁)) = x₁ 164.91/58.91
POL(c13(x₁)) = x₁ 164.91/58.91
POL(c14(x₁)) = x₁ 164.91/58.91
POL(c15(x₁)) = x₁ 164.91/58.91
POL(c16(x₁)) = x₁ 164.91/58.91
POL(c8(x₁)) = x₁ 164.91/58.91
POL(c9(x₁)) = x₁ 164.91/58.91
POL(mark(x₁)) = [1] + x₁ 164.91/58.91
POL(ok(x₁)) = x₁

164.91/58.91

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 164.91/58.91
active(sel(0, cons(z0, z1))) → mark(z0) 164.91/58.91
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 164.91/58.91
active(from(z0)) → from(active(z0)) 164.91/58.91
active(cons(z0, z1)) → cons(active(z0), z1) 164.91/58.91
active(s(z0)) → s(active(z0)) 165.29/58.91
active(sel(z0, z1)) → sel(active(z0), z1) 165.29/58.91
active(sel(z0, z1)) → sel(z0, active(z1)) 165.29/58.91
from(mark(z0)) → mark(from(z0)) 165.29/58.91
from(ok(z0)) → ok(from(z0)) 165.29/58.91
cons(mark(z0), z1) → mark(cons(z0, z1)) 165.29/58.91
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 165.29/58.91
s(mark(z0)) → mark(s(z0)) 165.29/58.91
s(ok(z0)) → ok(s(z0)) 165.29/58.91
sel(mark(z0), z1) → mark(sel(z0, z1)) 165.29/58.91
sel(z0, mark(z1)) → mark(sel(z0, z1)) 165.29/58.91
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 165.29/58.91
proper(from(z0)) → from(proper(z0)) 165.29/58.91
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 165.29/58.91
proper(s(z0)) → s(proper(z0)) 165.29/58.91
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 165.29/58.91
proper(0) → ok(0) 165.29/58.91
top(mark(z0)) → top(proper(z0)) 165.29/58.91
top(ok(z0)) → top(active(z0))

Tuples:

FROM(mark(z0)) → c8(FROM(z0)) 165.29/58.91
FROM(ok(z0)) → c9(FROM(z0)) 165.29/58.91
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 165.29/58.91
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 165.29/58.91
S(mark(z0)) → c12(S(z0)) 165.29/58.91
S(ok(z0)) → c13(S(z0)) 165.29/58.91
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 165.29/58.91
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 165.29/58.91
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))

S tuples:

FROM(ok(z0)) → c9(FROM(z0)) 165.29/58.91
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))

K tuples:

S(mark(z0)) → c12(S(z0)) 165.29/58.91
S(ok(z0)) → c13(S(z0)) 165.29/58.91
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 165.29/58.91
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 165.29/58.91
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1)) 165.29/58.91
FROM(mark(z0)) → c8(FROM(z0)) 165.29/58.91
CONS(mark(z0), z1) → c10(CONS(z0, z1))

Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

FROM, CONS, S, SEL

Compound Symbols:

c8, c9, c10, c11, c12, c13, c14, c15, c16

165.29/58.91

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

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

FROM(ok(z0)) → c9(FROM(z0))

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

FROM(mark(z0)) → c8(FROM(z0)) 165.29/58.91
FROM(ok(z0)) → c9(FROM(z0)) 165.29/58.91
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 165.29/58.91
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 165.29/58.91
S(mark(z0)) → c12(S(z0)) 165.29/58.91
S(ok(z0)) → c13(S(z0)) 165.29/58.91
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 165.29/58.91
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 165.29/58.91
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 165.29/58.91

POL(CONS(x₁, x₂)) = 0 165.29/58.91
POL(FROM(x₁)) = x₁² 165.29/58.91
POL(S(x₁)) = [3]x₁ + [3]x₁² 165.29/58.91
POL(SEL(x₁, x₂)) = x₁ + x₂ 165.29/58.91
POL(c10(x₁)) = x₁ 165.29/58.91
POL(c11(x₁)) = x₁ 165.29/58.91
POL(c12(x₁)) = x₁ 165.29/58.91
POL(c13(x₁)) = x₁ 165.29/58.91
POL(c14(x₁)) = x₁ 165.29/58.91
POL(c15(x₁)) = x₁ 165.29/58.91
POL(c16(x₁)) = x₁ 165.29/58.91
POL(c8(x₁)) = x₁ 165.29/58.91
POL(c9(x₁)) = x₁ 165.29/58.91
POL(mark(x₁)) = [3] + x₁ 165.29/58.91
POL(ok(x₁)) = [1] + x₁

165.29/58.91

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 165.29/58.91
active(sel(0, cons(z0, z1))) → mark(z0) 165.29/58.91
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 165.29/58.91
active(from(z0)) → from(active(z0)) 165.29/58.91
active(cons(z0, z1)) → cons(active(z0), z1) 165.29/58.91
active(s(z0)) → s(active(z0)) 165.29/58.91
active(sel(z0, z1)) → sel(active(z0), z1) 165.29/58.91
active(sel(z0, z1)) → sel(z0, active(z1)) 165.29/58.91
from(mark(z0)) → mark(from(z0)) 165.29/58.91
from(ok(z0)) → ok(from(z0)) 165.29/58.91
cons(mark(z0), z1) → mark(cons(z0, z1)) 165.29/58.91
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 165.29/58.91
s(mark(z0)) → mark(s(z0)) 165.29/58.91
s(ok(z0)) → ok(s(z0)) 165.29/58.91
sel(mark(z0), z1) → mark(sel(z0, z1)) 165.29/58.91
sel(z0, mark(z1)) → mark(sel(z0, z1)) 165.29/58.91
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 165.29/58.91
proper(from(z0)) → from(proper(z0)) 165.29/58.91
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 165.29/58.91
proper(s(z0)) → s(proper(z0)) 165.29/58.91
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 165.29/58.91
proper(0) → ok(0) 165.29/58.91
top(mark(z0)) → top(proper(z0)) 165.29/58.91
top(ok(z0)) → top(active(z0))

Tuples:

FROM(mark(z0)) → c8(FROM(z0)) 165.29/58.91
FROM(ok(z0)) → c9(FROM(z0)) 165.29/58.91
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 165.29/58.91
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 165.29/58.91
S(mark(z0)) → c12(S(z0)) 165.29/58.91
S(ok(z0)) → c13(S(z0)) 165.29/58.91
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 165.29/58.91
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 165.29/58.91
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))

S tuples:

CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))

K tuples:

S(mark(z0)) → c12(S(z0)) 165.29/58.91
S(ok(z0)) → c13(S(z0)) 165.29/58.91
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 165.29/58.91
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 165.29/58.91
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1)) 165.29/58.91
FROM(mark(z0)) → c8(FROM(z0)) 165.29/58.91
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 165.29/58.91
FROM(ok(z0)) → c9(FROM(z0))

Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

FROM, CONS, S, SEL

Compound Symbols:

c8, c9, c10, c11, c12, c13, c14, c15, c16

165.29/58.91

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

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

CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))

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

FROM(mark(z0)) → c8(FROM(z0)) 165.29/58.91
FROM(ok(z0)) → c9(FROM(z0)) 165.29/58.91
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 165.29/58.91
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 165.29/58.91
S(mark(z0)) → c12(S(z0)) 165.29/58.91
S(ok(z0)) → c13(S(z0)) 165.29/58.91
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 165.29/58.91
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 165.29/58.91
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 165.29/58.91

POL(CONS(x₁, x₂)) = [2]x₁ 165.29/58.91
POL(FROM(x₁)) = [5]x₁ 165.29/58.91
POL(S(x₁)) = [3]x₁ 165.29/58.91
POL(SEL(x₁, x₂)) = [5]x₁ + [3]x₂ 165.29/58.91
POL(c10(x₁)) = x₁ 165.29/58.91
POL(c11(x₁)) = x₁ 165.29/58.91
POL(c12(x₁)) = x₁ 165.29/58.91
POL(c13(x₁)) = x₁ 165.29/58.91
POL(c14(x₁)) = x₁ 165.29/58.91
POL(c15(x₁)) = x₁ 165.29/58.91
POL(c16(x₁)) = x₁ 165.29/58.91
POL(c8(x₁)) = x₁ 165.29/58.91
POL(c9(x₁)) = x₁ 165.29/58.91
POL(mark(x₁)) = x₁ 165.29/58.91
POL(ok(x₁)) = [1] + x₁

165.29/58.91

(50) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 165.29/58.91
active(sel(0, cons(z0, z1))) → mark(z0) 165.29/58.91
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 165.29/58.91
active(from(z0)) → from(active(z0)) 165.29/58.91
active(cons(z0, z1)) → cons(active(z0), z1) 165.29/58.91
active(s(z0)) → s(active(z0)) 165.29/58.91
active(sel(z0, z1)) → sel(active(z0), z1) 165.29/58.91
active(sel(z0, z1)) → sel(z0, active(z1)) 165.29/58.91
from(mark(z0)) → mark(from(z0)) 165.29/58.91
from(ok(z0)) → ok(from(z0)) 165.29/58.91
cons(mark(z0), z1) → mark(cons(z0, z1)) 165.29/58.91
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 165.29/58.91
s(mark(z0)) → mark(s(z0)) 165.29/58.91
s(ok(z0)) → ok(s(z0)) 165.29/58.91
sel(mark(z0), z1) → mark(sel(z0, z1)) 165.29/58.91
sel(z0, mark(z1)) → mark(sel(z0, z1)) 165.29/58.91
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 165.29/58.91
proper(from(z0)) → from(proper(z0)) 165.29/58.91
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 165.29/58.91
proper(s(z0)) → s(proper(z0)) 165.29/58.91
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 165.29/58.91
proper(0) → ok(0) 165.29/58.91
top(mark(z0)) → top(proper(z0)) 165.29/58.91
top(ok(z0)) → top(active(z0))

Tuples:

FROM(mark(z0)) → c8(FROM(z0)) 165.29/58.91
FROM(ok(z0)) → c9(FROM(z0)) 165.29/58.91
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 165.29/58.91
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 165.29/58.91
S(mark(z0)) → c12(S(z0)) 165.29/58.91
S(ok(z0)) → c13(S(z0)) 165.29/58.91
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 165.29/58.91
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 165.29/58.91
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1))

S tuples:none
K tuples:

S(mark(z0)) → c12(S(z0)) 165.29/58.91
S(ok(z0)) → c13(S(z0)) 165.29/58.91
SEL(mark(z0), z1) → c14(SEL(z0, z1)) 165.29/58.91
SEL(z0, mark(z1)) → c15(SEL(z0, z1)) 165.29/58.91
SEL(ok(z0), ok(z1)) → c16(SEL(z0, z1)) 165.29/58.91
FROM(mark(z0)) → c8(FROM(z0)) 165.29/58.91
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 165.29/58.91
FROM(ok(z0)) → c9(FROM(z0)) 165.29/58.91
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))

Defined Rule Symbols:

active, from, cons, s, sel, proper, top

Defined Pair Symbols:

FROM, CONS, S, SEL

Compound Symbols:

c8, c9, c10, c11, c12, c13, c14, c15, c16

165.29/58.91

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

The set S is empty

165.29/58.91

(52) BOUNDS(O(1), O(1))

165.29/58.91