YES(O(1), O(n^3)) 157.46/57.80 YES(O(1), O(n^3)) 158.08/57.93 158.08/57.93 158.08/57.93 158.08/57.93 158.08/57.93 158.08/57.93 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 158.08/57.93 158.08/57.93 158.08/57.93
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158.08/57.93
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^3)).

0 CpxTRS
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↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
158.08/57.93 
↳2 CdtProblem
158.08/57.93 
↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
158.08/57.93 
↳4 CdtProblem
158.08/57.93 
↳5 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
158.08/57.93 
↳6 CdtProblem
158.08/57.93 
↳7 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
158.08/57.93 
↳8 CdtProblem
158.08/57.93 
↳9 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
158.08/57.93 
↳10 CdtProblem
158.08/57.93 
↳11 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
158.08/57.93 
↳12 CdtProblem
158.08/57.93 
↳13 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
158.08/57.93 
↳14 CdtProblem
158.08/57.93 
↳15 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
158.08/57.93 
↳16 CdtProblem
158.08/57.93 
↳17 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
158.08/57.93 
↳18 CdtProblem
158.08/57.93 
↳19 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
158.08/57.93 
↳20 CdtProblem
158.08/57.93 
↳21 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
158.08/57.93 
↳22 CdtProblem
158.08/57.93 
↳23 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
158.08/57.93 
↳24 CdtProblem
158.08/57.93 
↳25 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
158.08/57.93 
↳26 CdtProblem
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↳27 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
158.08/57.93 
↳28 CdtProblem
158.08/57.93 
↳29 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
158.08/57.93 
↳30 CdtProblem
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↳31 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
158.08/57.93 
↳32 CdtProblem
158.08/57.93 
↳33 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
158.08/57.93 
↳34 CdtProblem
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↳35 CdtUnreachableProof (⇔)
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↳36 CdtProblem
158.08/57.93 
↳37 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
158.08/57.93 
↳38 CdtProblem
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↳39 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
158.08/57.93 
↳40 CdtProblem
158.08/57.93 
↳41 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
158.08/57.93 
↳42 CdtProblem
158.08/57.93 
↳43 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^3))))
158.08/57.93 
↳44 CdtProblem
158.08/57.93 
↳45 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
158.08/57.93 
↳46 CdtProblem
158.08/57.93 
↳47 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
158.08/57.93 
↳48 CdtProblem
158.08/57.93 
↳49 SIsEmptyProof (BOTH BOUNDS(ID, ID))
158.08/57.93 
↳50 BOUNDS(O(1), O(1))
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158.08/57.93

(0) Obligation:

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

active(from(X)) → mark(cons(X, from(s(X)))) 158.08/57.93
active(after(0, XS)) → mark(XS) 158.08/57.93
active(after(s(N), cons(X, XS))) → mark(after(N, XS)) 158.08/57.93
active(from(X)) → from(active(X)) 158.08/57.93
active(cons(X1, X2)) → cons(active(X1), X2) 158.08/57.93
active(s(X)) → s(active(X)) 158.08/57.93
active(after(X1, X2)) → after(active(X1), X2) 158.08/57.93
active(after(X1, X2)) → after(X1, active(X2)) 158.08/57.93
from(mark(X)) → mark(from(X)) 158.08/57.93
cons(mark(X1), X2) → mark(cons(X1, X2)) 158.08/57.93
s(mark(X)) → mark(s(X)) 158.08/57.93
after(mark(X1), X2) → mark(after(X1, X2)) 158.08/57.93
after(X1, mark(X2)) → mark(after(X1, X2)) 158.08/57.93
proper(from(X)) → from(proper(X)) 158.08/57.93
proper(cons(X1, X2)) → cons(proper(X1), proper(X2)) 158.08/57.93
proper(s(X)) → s(proper(X)) 158.08/57.93
proper(after(X1, X2)) → after(proper(X1), proper(X2)) 158.08/57.93
proper(0) → ok(0) 158.08/57.93
from(ok(X)) → ok(from(X)) 158.08/57.93
cons(ok(X1), ok(X2)) → ok(cons(X1, X2)) 158.08/57.93
s(ok(X)) → ok(s(X)) 158.08/57.93
after(ok(X1), ok(X2)) → ok(after(X1, X2)) 158.08/57.93
top(mark(X)) → top(proper(X)) 158.08/57.93
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST
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158.08/57.93

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

Converted CpxTRS to CDT
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158.08/57.93

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 158.08/57.93
active(after(0, z0)) → mark(z0) 158.08/57.93
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2)) 158.08/57.93
active(from(z0)) → from(active(z0)) 158.08/57.93
active(cons(z0, z1)) → cons(active(z0), z1) 158.08/57.93
active(s(z0)) → s(active(z0)) 158.08/57.93
active(after(z0, z1)) → after(active(z0), z1) 158.08/57.93
active(after(z0, z1)) → after(z0, active(z1)) 158.08/57.93
from(mark(z0)) → mark(from(z0)) 158.08/57.93
from(ok(z0)) → ok(from(z0)) 158.08/57.93
cons(mark(z0), z1) → mark(cons(z0, z1)) 158.08/57.93
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 158.08/57.93
s(mark(z0)) → mark(s(z0)) 158.08/57.93
s(ok(z0)) → ok(s(z0)) 158.08/57.93
after(mark(z0), z1) → mark(after(z0, z1)) 158.08/57.93
after(z0, mark(z1)) → mark(after(z0, z1)) 158.08/57.93
after(ok(z0), ok(z1)) → ok(after(z0, z1)) 158.08/57.93
proper(from(z0)) → from(proper(z0)) 158.08/57.93
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 158.08/57.93
proper(s(z0)) → s(proper(z0)) 158.08/57.93
proper(after(z0, z1)) → after(proper(z0), proper(z1)) 158.08/57.93
proper(0) → ok(0) 158.08/57.93
top(mark(z0)) → top(proper(z0)) 158.08/57.93
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(from(z0)) → c(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0)) 158.08/57.93
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2)) 158.08/57.93
ACTIVE(from(z0)) → c3(FROM(active(z0)), ACTIVE(z0)) 158.08/57.93
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0)) 158.08/57.93
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0)) 158.08/57.93
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0)) 158.08/57.93
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1)) 158.08/57.93
FROM(mark(z0)) → c8(FROM(z0)) 158.08/57.93
FROM(ok(z0)) → c9(FROM(z0)) 158.08/57.93
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.08/57.93
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.08/57.93
S(mark(z0)) → c12(S(z0)) 158.08/57.93
S(ok(z0)) → c13(S(z0)) 158.08/57.93
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.08/57.93
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.08/57.93
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1)) 158.08/57.93
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0)) 158.08/57.93
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 158.08/57.93
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0)) 158.08/57.93
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 158.08/57.93
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 158.08/57.93
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)) 158.08/57.93
ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2)) 158.08/57.93
ACTIVE(from(z0)) → c3(FROM(active(z0)), ACTIVE(z0)) 158.08/57.93
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0)) 158.08/57.93
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0)) 158.08/57.93
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0)) 158.08/57.93
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1)) 158.08/57.93
FROM(mark(z0)) → c8(FROM(z0)) 158.08/57.93
FROM(ok(z0)) → c9(FROM(z0)) 158.08/57.93
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.08/57.93
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.08/57.93
S(mark(z0)) → c12(S(z0)) 158.08/57.93
S(ok(z0)) → c13(S(z0)) 158.08/57.93
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.08/57.93
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.08/57.93
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1)) 158.08/57.93
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0)) 158.08/57.93
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 158.08/57.93
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0)) 158.08/57.93
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 158.08/57.93
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 158.08/57.93
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, after, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, AFTER, 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

158.08/57.93
158.08/57.93

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

Removed 2 trailing tuple parts
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158.08/57.93

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 158.08/57.93
active(after(0, z0)) → mark(z0) 158.08/57.93
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2)) 158.08/57.93
active(from(z0)) → from(active(z0)) 158.08/57.93
active(cons(z0, z1)) → cons(active(z0), z1) 158.08/57.93
active(s(z0)) → s(active(z0)) 158.08/57.93
active(after(z0, z1)) → after(active(z0), z1) 158.08/57.93
active(after(z0, z1)) → after(z0, active(z1)) 158.08/57.93
from(mark(z0)) → mark(from(z0)) 158.08/57.93
from(ok(z0)) → ok(from(z0)) 158.08/57.93
cons(mark(z0), z1) → mark(cons(z0, z1)) 158.08/57.93
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 158.08/57.93
s(mark(z0)) → mark(s(z0)) 158.08/57.93
s(ok(z0)) → ok(s(z0)) 158.08/57.93
after(mark(z0), z1) → mark(after(z0, z1)) 158.08/57.93
after(z0, mark(z1)) → mark(after(z0, z1)) 158.08/57.93
after(ok(z0), ok(z1)) → ok(after(z0, z1)) 158.08/57.93
proper(from(z0)) → from(proper(z0)) 158.08/57.93
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 158.08/57.93
proper(s(z0)) → s(proper(z0)) 158.08/57.93
proper(after(z0, z1)) → after(proper(z0), proper(z1)) 158.08/57.93
proper(0) → ok(0) 158.08/57.93
top(mark(z0)) → top(proper(z0)) 158.08/57.93
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2)) 158.08/57.93
ACTIVE(from(z0)) → c3(FROM(active(z0)), ACTIVE(z0)) 158.08/57.93
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0)) 158.08/57.93
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0)) 158.08/57.93
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0)) 158.08/57.93
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1)) 158.08/57.93
FROM(mark(z0)) → c8(FROM(z0)) 158.08/57.93
FROM(ok(z0)) → c9(FROM(z0)) 158.08/57.93
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.08/57.93
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.08/57.93
S(mark(z0)) → c12(S(z0)) 158.08/57.93
S(ok(z0)) → c13(S(z0)) 158.08/57.93
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.08/57.93
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.08/57.93
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1)) 158.08/57.93
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0)) 158.08/57.93
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 158.08/57.93
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0)) 158.08/57.93
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 158.08/57.93
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 158.08/57.93
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 158.08/57.93
ACTIVE(from(z0)) → c(S(z0))
S tuples:

ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2)) 158.08/57.94
ACTIVE(from(z0)) → c3(FROM(active(z0)), ACTIVE(z0)) 158.08/57.94
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0)) 158.08/57.94
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0)) 158.08/57.94
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0)) 158.08/57.94
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1)) 158.08/57.94
FROM(mark(z0)) → c8(FROM(z0)) 158.08/57.94
FROM(ok(z0)) → c9(FROM(z0)) 158.08/57.94
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.08/57.94
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.08/57.94
S(mark(z0)) → c12(S(z0)) 158.08/57.94
S(ok(z0)) → c13(S(z0)) 158.08/57.94
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.08/57.94
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.08/57.94
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1)) 158.08/57.94
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0)) 158.08/57.94
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 158.08/57.94
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0)) 158.08/57.94
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 158.08/57.94
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 158.08/57.94
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 158.08/57.94
ACTIVE(from(z0)) → c(S(z0))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, after, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, AFTER, 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

158.08/57.94
158.08/57.94

(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))) 158.08/57.94
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0))) 158.08/57.94
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.08/57.94
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 158.08/57.94
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.08/57.94
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 158.08/57.94
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.08/57.94
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
158.08/57.94
158.08/57.94

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 158.08/57.94
active(after(0, z0)) → mark(z0) 158.08/57.94
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2)) 158.08/57.94
active(from(z0)) → from(active(z0)) 158.08/57.94
active(cons(z0, z1)) → cons(active(z0), z1) 158.08/57.94
active(s(z0)) → s(active(z0)) 158.08/57.94
active(after(z0, z1)) → after(active(z0), z1) 158.08/57.94
active(after(z0, z1)) → after(z0, active(z1)) 158.08/57.94
from(mark(z0)) → mark(from(z0)) 158.08/57.94
from(ok(z0)) → ok(from(z0)) 158.08/57.94
cons(mark(z0), z1) → mark(cons(z0, z1)) 158.08/57.94
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 158.08/57.94
s(mark(z0)) → mark(s(z0)) 158.08/57.94
s(ok(z0)) → ok(s(z0)) 158.08/57.94
after(mark(z0), z1) → mark(after(z0, z1)) 158.08/57.94
after(z0, mark(z1)) → mark(after(z0, z1)) 158.08/57.94
after(ok(z0), ok(z1)) → ok(after(z0, z1)) 158.08/57.94
proper(from(z0)) → from(proper(z0)) 158.08/57.94
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 158.08/57.94
proper(s(z0)) → s(proper(z0)) 158.08/57.94
proper(after(z0, z1)) → after(proper(z0), proper(z1)) 158.08/57.94
proper(0) → ok(0) 158.08/57.94
top(mark(z0)) → top(proper(z0)) 158.08/57.94
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2)) 158.08/57.94
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0)) 158.08/57.94
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0)) 158.08/57.94
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0)) 158.08/57.94
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1)) 158.08/57.94
FROM(mark(z0)) → c8(FROM(z0)) 158.08/57.94
FROM(ok(z0)) → c9(FROM(z0)) 158.08/57.94
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.08/57.94
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.08/57.94
S(mark(z0)) → c12(S(z0)) 158.08/57.94
S(ok(z0)) → c13(S(z0)) 158.08/57.94
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.08/57.94
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.08/57.94
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1)) 158.08/57.94
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0)) 158.08/57.94
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 158.08/57.94
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0)) 158.08/57.94
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 158.08/57.94
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 158.08/57.94
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 158.08/57.94
ACTIVE(from(z0)) → c(S(z0)) 158.08/57.94
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.08/57.94
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0))) 158.08/57.94
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.08/57.94
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 158.08/57.94
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.08/57.94
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 158.08/57.94
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.08/57.94
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
S tuples:

ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2)) 158.08/57.94
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0)) 158.08/57.94
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0)) 158.08/57.94
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0)) 158.08/57.94
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1)) 158.08/57.94
FROM(mark(z0)) → c8(FROM(z0)) 158.08/57.94
FROM(ok(z0)) → c9(FROM(z0)) 158.08/57.94
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.08/57.94
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.08/58.00
S(mark(z0)) → c12(S(z0)) 158.08/58.00
S(ok(z0)) → c13(S(z0)) 158.08/58.00
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.08/58.00
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.08/58.00
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1)) 158.08/58.00
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0)) 158.08/58.00
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 158.08/58.00
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0)) 158.08/58.00
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 158.08/58.00
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 158.08/58.00
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 158.08/58.00
ACTIVE(from(z0)) → c(S(z0)) 158.08/58.00
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.08/58.00
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0))) 158.08/58.00
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.08/58.00
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 158.08/58.00
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.08/58.00
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 158.08/58.00
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.08/58.00
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, after, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, AFTER, 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

158.08/58.00
158.08/58.00

(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))) 158.08/58.00
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0))) 158.08/58.00
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2)))) 158.08/58.00
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 158.08/58.00
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 158.08/58.00
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 158.08/58.00
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1))) 158.08/58.00
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
158.08/58.00
158.08/58.00

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 158.08/58.00
active(after(0, z0)) → mark(z0) 158.08/58.00
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2)) 158.08/58.00
active(from(z0)) → from(active(z0)) 158.08/58.00
active(cons(z0, z1)) → cons(active(z0), z1) 158.08/58.00
active(s(z0)) → s(active(z0)) 158.08/58.00
active(after(z0, z1)) → after(active(z0), z1) 158.08/58.00
active(after(z0, z1)) → after(z0, active(z1)) 158.08/58.00
from(mark(z0)) → mark(from(z0)) 158.08/58.00
from(ok(z0)) → ok(from(z0)) 158.08/58.00
cons(mark(z0), z1) → mark(cons(z0, z1)) 158.08/58.00
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 158.08/58.00
s(mark(z0)) → mark(s(z0)) 158.08/58.00
s(ok(z0)) → ok(s(z0)) 158.08/58.00
after(mark(z0), z1) → mark(after(z0, z1)) 158.08/58.00
after(z0, mark(z1)) → mark(after(z0, z1)) 158.08/58.00
after(ok(z0), ok(z1)) → ok(after(z0, z1)) 158.08/58.00
proper(from(z0)) → from(proper(z0)) 158.08/58.00
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 158.08/58.00
proper(s(z0)) → s(proper(z0)) 158.08/58.00
proper(after(z0, z1)) → after(proper(z0), proper(z1)) 158.08/58.00
proper(0) → ok(0) 158.08/58.00
top(mark(z0)) → top(proper(z0)) 158.08/58.00
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2)) 158.08/58.00
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0)) 158.08/58.00
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0)) 158.08/58.00
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1)) 158.08/58.00
FROM(mark(z0)) → c8(FROM(z0)) 158.08/58.00
FROM(ok(z0)) → c9(FROM(z0)) 158.08/58.00
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.08/58.00
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.08/58.00
S(mark(z0)) → c12(S(z0)) 158.08/58.00
S(ok(z0)) → c13(S(z0)) 158.08/58.00
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.08/58.00
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.08/58.00
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1)) 158.08/58.00
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0)) 158.08/58.00
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 158.08/58.00
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0)) 158.08/58.00
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 158.08/58.00
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 158.08/58.00
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 158.08/58.00
ACTIVE(from(z0)) → c(S(z0)) 158.08/58.00
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.08/58.00
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0))) 158.08/58.00
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.08/58.00
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 158.08/58.00
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.08/58.00
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 158.08/58.00
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.08/58.00
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.08/58.00
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 158.08/58.00
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0))) 158.08/58.00
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2)))) 158.08/58.00
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 158.08/58.00
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 158.08/58.00
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 158.08/58.00
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1))) 158.08/58.00
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
S tuples:

ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2)) 158.08/58.00
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0)) 158.08/58.00
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0)) 158.08/58.00
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1)) 158.08/58.00
FROM(mark(z0)) → c8(FROM(z0)) 158.08/58.00
FROM(ok(z0)) → c9(FROM(z0)) 158.08/58.00
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.08/58.00
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.08/58.00
S(mark(z0)) → c12(S(z0)) 158.08/58.00
S(ok(z0)) → c13(S(z0)) 158.08/58.00
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.08/58.00
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.08/58.00
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1)) 158.08/58.00
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0)) 158.08/58.00
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 158.08/58.00
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0)) 158.47/58.08
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 158.47/58.08
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 158.47/58.08
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 158.47/58.08
ACTIVE(from(z0)) → c(S(z0)) 158.47/58.08
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.47/58.08
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0))) 158.47/58.08
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.47/58.08
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 158.47/58.08
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.47/58.08
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 158.47/58.08
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.47/58.08
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.47/58.08
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 158.47/58.08
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0))) 158.47/58.08
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2)))) 158.47/58.08
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 158.47/58.08
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 158.47/58.08
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 158.47/58.08
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1))) 158.47/58.08
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, after, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, AFTER, 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

158.47/58.08
158.47/58.08

(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))) 158.47/58.08
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0))) 158.47/58.08
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.47/58.08
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 158.47/58.08
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.47/58.08
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 158.47/58.08
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.47/58.08
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
158.47/58.08
158.47/58.08

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 158.47/58.08
active(after(0, z0)) → mark(z0) 158.47/58.08
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2)) 158.47/58.08
active(from(z0)) → from(active(z0)) 158.47/58.08
active(cons(z0, z1)) → cons(active(z0), z1) 158.47/58.08
active(s(z0)) → s(active(z0)) 158.47/58.08
active(after(z0, z1)) → after(active(z0), z1) 158.47/58.08
active(after(z0, z1)) → after(z0, active(z1)) 158.47/58.08
from(mark(z0)) → mark(from(z0)) 158.47/58.08
from(ok(z0)) → ok(from(z0)) 158.47/58.08
cons(mark(z0), z1) → mark(cons(z0, z1)) 158.47/58.08
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 158.47/58.08
s(mark(z0)) → mark(s(z0)) 158.47/58.08
s(ok(z0)) → ok(s(z0)) 158.47/58.08
after(mark(z0), z1) → mark(after(z0, z1)) 158.47/58.08
after(z0, mark(z1)) → mark(after(z0, z1)) 158.47/58.08
after(ok(z0), ok(z1)) → ok(after(z0, z1)) 158.47/58.08
proper(from(z0)) → from(proper(z0)) 158.47/58.08
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 158.47/58.08
proper(s(z0)) → s(proper(z0)) 158.47/58.08
proper(after(z0, z1)) → after(proper(z0), proper(z1)) 158.47/58.08
proper(0) → ok(0) 158.47/58.08
top(mark(z0)) → top(proper(z0)) 158.47/58.08
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2)) 158.47/58.08
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0)) 158.47/58.08
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1)) 158.47/58.08
FROM(mark(z0)) → c8(FROM(z0)) 158.47/58.08
FROM(ok(z0)) → c9(FROM(z0)) 158.47/58.08
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.47/58.08
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.47/58.08
S(mark(z0)) → c12(S(z0)) 158.47/58.08
S(ok(z0)) → c13(S(z0)) 158.47/58.08
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.47/58.08
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.47/58.08
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1)) 158.47/58.08
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0)) 158.47/58.08
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 158.47/58.08
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0)) 158.47/58.08
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 158.47/58.08
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 158.47/58.08
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 158.47/58.08
ACTIVE(from(z0)) → c(S(z0)) 158.47/58.08
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.47/58.08
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0))) 158.47/58.08
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.47/58.08
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 158.47/58.08
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.47/58.08
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 158.47/58.08
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.47/58.08
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.47/58.08
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 158.47/58.08
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0))) 158.47/58.08
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2)))) 158.47/58.08
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 158.47/58.08
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 158.47/58.08
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 158.47/58.08
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1))) 158.47/58.08
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1))) 158.47/58.08
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.47/58.08
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0))) 158.47/58.08
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.47/58.08
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 158.47/58.08
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.47/58.08
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 158.47/58.08
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.47/58.08
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
S tuples:

ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2)) 158.47/58.08
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0)) 158.47/58.08
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1)) 158.47/58.08
FROM(mark(z0)) → c8(FROM(z0)) 158.47/58.08
FROM(ok(z0)) → c9(FROM(z0)) 158.47/58.08
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.47/58.08
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.47/58.08
S(mark(z0)) → c12(S(z0)) 158.47/58.08
S(ok(z0)) → c13(S(z0)) 158.47/58.08
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.47/58.08
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.47/58.08
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1)) 158.47/58.08
PROPER(from(z0)) → c17(FROM(proper(z0)), PROPER(z0)) 158.47/58.08
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 158.47/58.08
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0)) 158.47/58.08
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 158.47/58.08
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 158.47/58.08
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 158.47/58.08
ACTIVE(from(z0)) → c(S(z0)) 158.47/58.08
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.47/58.08
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0))) 158.47/58.08
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.47/58.08
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 158.47/58.08
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.47/58.08
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 158.47/58.08
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.47/58.08
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.47/58.08
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 158.47/58.08
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0))) 158.47/58.08
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2)))) 158.47/58.08
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 158.47/58.08
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 158.47/58.08
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 158.47/58.08
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1))) 158.47/58.08
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1))) 158.47/58.08
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.47/58.08
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0))) 158.47/58.08
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.47/58.08
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 158.47/58.08
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.47/58.08
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 158.47/58.08
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.47/58.08
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, after, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, AFTER, 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

158.47/58.08
158.47/58.08

(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))) 158.47/58.08
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.47/58.08
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 158.47/58.08
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.47/58.08
PROPER(from(0)) → c17(FROM(ok(0)), PROPER(0))
158.47/58.08
158.47/58.08

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 158.47/58.08
active(after(0, z0)) → mark(z0) 158.47/58.08
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2)) 158.47/58.09
active(from(z0)) → from(active(z0)) 158.47/58.09
active(cons(z0, z1)) → cons(active(z0), z1) 158.47/58.09
active(s(z0)) → s(active(z0)) 158.47/58.09
active(after(z0, z1)) → after(active(z0), z1) 158.47/58.09
active(after(z0, z1)) → after(z0, active(z1)) 158.47/58.09
from(mark(z0)) → mark(from(z0)) 158.47/58.09
from(ok(z0)) → ok(from(z0)) 158.47/58.09
cons(mark(z0), z1) → mark(cons(z0, z1)) 158.47/58.09
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 158.47/58.09
s(mark(z0)) → mark(s(z0)) 158.47/58.09
s(ok(z0)) → ok(s(z0)) 158.47/58.09
after(mark(z0), z1) → mark(after(z0, z1)) 158.47/58.09
after(z0, mark(z1)) → mark(after(z0, z1)) 158.47/58.09
after(ok(z0), ok(z1)) → ok(after(z0, z1)) 158.47/58.09
proper(from(z0)) → from(proper(z0)) 158.47/58.09
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 158.47/58.09
proper(s(z0)) → s(proper(z0)) 158.47/58.09
proper(after(z0, z1)) → after(proper(z0), proper(z1)) 158.47/58.09
proper(0) → ok(0) 158.47/58.09
top(mark(z0)) → top(proper(z0)) 158.47/58.09
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2)) 158.47/58.09
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0)) 158.47/58.09
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1)) 158.47/58.09
FROM(mark(z0)) → c8(FROM(z0)) 158.47/58.09
FROM(ok(z0)) → c9(FROM(z0)) 158.47/58.09
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.47/58.09
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.47/58.09
S(mark(z0)) → c12(S(z0)) 158.47/58.09
S(ok(z0)) → c13(S(z0)) 158.47/58.09
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.47/58.09
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.47/58.09
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1)) 158.47/58.09
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 158.47/58.09
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0)) 158.47/58.09
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 158.47/58.09
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 158.47/58.09
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 158.47/58.09
ACTIVE(from(z0)) → c(S(z0)) 158.47/58.09
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.47/58.09
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0))) 158.47/58.09
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.47/58.09
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 158.47/58.09
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.47/58.09
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 158.47/58.09
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.47/58.09
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.47/58.09
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 158.47/58.09
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0))) 158.47/58.09
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2)))) 158.47/58.09
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 158.47/58.09
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 158.47/58.09
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 158.47/58.09
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1))) 158.47/58.09
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1))) 158.47/58.09
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.47/58.09
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0))) 158.47/58.09
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.47/58.09
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 158.47/58.09
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.47/58.09
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 158.47/58.09
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.47/58.09
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.47/58.09
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 158.47/58.09
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.47/58.09
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 158.47/58.09
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.47/58.09
PROPER(from(0)) → c17(FROM(ok(0)), PROPER(0))
S tuples:

ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2)) 158.47/58.09
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0)) 158.47/58.09
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1)) 158.47/58.09
FROM(mark(z0)) → c8(FROM(z0)) 158.47/58.09
FROM(ok(z0)) → c9(FROM(z0)) 158.47/58.09
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.47/58.09
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.47/58.09
S(mark(z0)) → c12(S(z0)) 158.47/58.09
S(ok(z0)) → c13(S(z0)) 158.47/58.09
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.47/58.09
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.47/58.09
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1)) 158.47/58.09
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 158.47/58.09
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0)) 158.47/58.09
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 158.47/58.09
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 158.47/58.09
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 158.47/58.09
ACTIVE(from(z0)) → c(S(z0)) 158.47/58.09
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.47/58.09
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0))) 158.47/58.09
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.47/58.09
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 158.47/58.09
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.47/58.09
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 158.47/58.09
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.47/58.09
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.47/58.09
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 158.47/58.09
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0))) 158.47/58.09
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2)))) 158.47/58.09
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 158.47/58.09
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 158.47/58.09
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 158.47/58.09
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1))) 158.47/58.09
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1))) 158.47/58.09
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.47/58.09
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0))) 158.47/58.09
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.47/58.09
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 158.47/58.09
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.47/58.09
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 158.47/58.09
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.47/58.09
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.47/58.09
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 158.47/58.09
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.47/58.09
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 158.47/58.09
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.47/58.09
PROPER(from(0)) → c17(FROM(ok(0)), PROPER(0))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, after, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, AFTER, 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

158.47/58.09
158.47/58.09

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

Removed 1 trailing tuple parts
158.47/58.09
158.47/58.09

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 158.47/58.09
active(after(0, z0)) → mark(z0) 158.47/58.09
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2)) 158.47/58.09
active(from(z0)) → from(active(z0)) 158.47/58.09
active(cons(z0, z1)) → cons(active(z0), z1) 158.47/58.09
active(s(z0)) → s(active(z0)) 158.47/58.09
active(after(z0, z1)) → after(active(z0), z1) 158.47/58.09
active(after(z0, z1)) → after(z0, active(z1)) 158.47/58.09
from(mark(z0)) → mark(from(z0)) 158.47/58.09
from(ok(z0)) → ok(from(z0)) 158.47/58.09
cons(mark(z0), z1) → mark(cons(z0, z1)) 158.47/58.09
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 158.47/58.09
s(mark(z0)) → mark(s(z0)) 158.47/58.09
s(ok(z0)) → ok(s(z0)) 158.47/58.09
after(mark(z0), z1) → mark(after(z0, z1)) 158.47/58.09
after(z0, mark(z1)) → mark(after(z0, z1)) 158.47/58.09
after(ok(z0), ok(z1)) → ok(after(z0, z1)) 158.47/58.09
proper(from(z0)) → from(proper(z0)) 158.47/58.09
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 158.47/58.09
proper(s(z0)) → s(proper(z0)) 158.47/58.09
proper(after(z0, z1)) → after(proper(z0), proper(z1)) 158.47/58.09
proper(0) → ok(0) 158.47/58.09
top(mark(z0)) → top(proper(z0)) 158.47/58.09
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2)) 158.47/58.09
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0)) 158.47/58.09
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1)) 158.47/58.09
FROM(mark(z0)) → c8(FROM(z0)) 158.47/58.09
FROM(ok(z0)) → c9(FROM(z0)) 158.47/58.09
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.47/58.09
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.47/58.09
S(mark(z0)) → c12(S(z0)) 158.47/58.09
S(ok(z0)) → c13(S(z0)) 158.47/58.09
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.47/58.09
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.47/58.09
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1)) 158.47/58.09
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 158.47/58.09
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0)) 158.47/58.09
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 158.47/58.09
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 158.47/58.09
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 158.47/58.09
ACTIVE(from(z0)) → c(S(z0)) 158.47/58.09
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.47/58.09
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0))) 158.47/58.09
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.47/58.09
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 158.47/58.09
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.47/58.09
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 158.47/58.09
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.47/58.09
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.47/58.09
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 158.47/58.09
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0))) 158.47/58.09
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2)))) 158.47/58.09
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 158.47/58.09
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 158.47/58.09
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 158.47/58.09
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1))) 158.47/58.09
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1))) 158.47/58.09
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.47/58.09
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0))) 158.47/58.09
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.47/58.09
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 158.47/58.09
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.47/58.09
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 158.47/58.09
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.47/58.09
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.47/58.09
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 158.47/58.09
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.47/58.09
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 158.47/58.09
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.47/58.09
PROPER(from(0)) → c17(FROM(ok(0)))
S tuples:

ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2)) 158.47/58.09
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0)) 158.47/58.09
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1)) 158.47/58.09
FROM(mark(z0)) → c8(FROM(z0)) 158.47/58.09
FROM(ok(z0)) → c9(FROM(z0)) 158.47/58.09
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.47/58.09
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.47/58.09
S(mark(z0)) → c12(S(z0)) 158.47/58.09
S(ok(z0)) → c13(S(z0)) 158.47/58.09
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.47/58.09
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.47/58.09
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1)) 158.47/58.09
PROPER(cons(z0, z1)) → c18(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 158.47/58.09
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0)) 158.47/58.09
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 158.47/58.09
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 158.47/58.09
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 158.47/58.09
ACTIVE(from(z0)) → c(S(z0)) 158.47/58.09
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.47/58.09
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0))) 158.47/58.09
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.47/58.09
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 158.47/58.09
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.47/58.09
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 158.47/58.09
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.47/58.09
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.47/58.09
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 158.47/58.09
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0))) 158.47/58.09
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2)))) 158.47/58.09
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 158.47/58.09
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 158.47/58.09
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 158.47/58.09
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1))) 158.47/58.09
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1))) 158.47/58.09
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.47/58.09
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0))) 158.47/58.09
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.47/58.09
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 158.47/58.09
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.47/58.09
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 158.47/58.09
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.47/58.09
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.47/58.09
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 158.47/58.09
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.47/58.09
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 158.47/58.09
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.47/58.09
PROPER(from(0)) → c17(FROM(ok(0)))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, after, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, AFTER, 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

158.47/58.09
158.47/58.09

(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))) 158.47/58.09
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 158.47/58.09
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 158.47/58.09
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1))) 158.47/58.09
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 158.47/58.09
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 158.47/58.09
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 158.47/58.09
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 158.47/58.09
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1)) 158.47/58.09
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
158.47/58.09
158.47/58.09

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 158.47/58.09
active(after(0, z0)) → mark(z0) 158.47/58.09
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2)) 158.47/58.09
active(from(z0)) → from(active(z0)) 158.47/58.09
active(cons(z0, z1)) → cons(active(z0), z1) 158.47/58.09
active(s(z0)) → s(active(z0)) 158.47/58.09
active(after(z0, z1)) → after(active(z0), z1) 158.47/58.09
active(after(z0, z1)) → after(z0, active(z1)) 158.47/58.09
from(mark(z0)) → mark(from(z0)) 158.47/58.09
from(ok(z0)) → ok(from(z0)) 158.47/58.09
cons(mark(z0), z1) → mark(cons(z0, z1)) 158.47/58.09
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 158.47/58.09
s(mark(z0)) → mark(s(z0)) 158.47/58.09
s(ok(z0)) → ok(s(z0)) 158.47/58.09
after(mark(z0), z1) → mark(after(z0, z1)) 158.47/58.09
after(z0, mark(z1)) → mark(after(z0, z1)) 158.47/58.09
after(ok(z0), ok(z1)) → ok(after(z0, z1)) 158.47/58.09
proper(from(z0)) → from(proper(z0)) 158.47/58.09
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 158.47/58.09
proper(s(z0)) → s(proper(z0)) 158.47/58.09
proper(after(z0, z1)) → after(proper(z0), proper(z1)) 158.47/58.09
proper(0) → ok(0) 158.47/58.09
top(mark(z0)) → top(proper(z0)) 158.47/58.09
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2)) 158.47/58.09
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0)) 158.47/58.09
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1)) 158.47/58.09
FROM(mark(z0)) → c8(FROM(z0)) 158.47/58.09
FROM(ok(z0)) → c9(FROM(z0)) 158.47/58.09
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.47/58.09
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.47/58.09
S(mark(z0)) → c12(S(z0)) 158.47/58.09
S(ok(z0)) → c13(S(z0)) 158.47/58.09
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.47/58.09
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.47/58.09
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1)) 158.47/58.09
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0)) 158.47/58.09
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 158.47/58.09
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 158.47/58.09
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 158.47/58.09
ACTIVE(from(z0)) → c(S(z0)) 158.47/58.09
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.47/58.09
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0))) 158.47/58.09
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.47/58.09
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 158.47/58.09
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.47/58.09
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 158.47/58.09
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.47/58.09
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.47/58.09
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 158.47/58.09
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0))) 158.47/58.09
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2)))) 158.47/58.09
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 158.47/58.09
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 158.47/58.09
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 158.47/58.09
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1))) 158.47/58.09
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1))) 158.47/58.09
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.47/58.09
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0))) 158.47/58.09
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.47/58.09
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 158.47/58.09
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.47/58.09
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 158.47/58.09
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.47/58.09
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.47/58.09
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 158.47/58.09
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.47/58.09
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 158.47/58.09
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.47/58.09
PROPER(from(0)) → c17(FROM(ok(0))) 158.47/58.09
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 158.47/58.09
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 158.47/58.09
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 158.47/58.09
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1))) 158.47/58.09
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 158.47/58.09
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 158.47/58.09
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 158.47/58.09
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 158.47/58.09
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1)) 158.47/58.09
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
S tuples:

ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2)) 158.47/58.09
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0)) 158.47/58.09
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1)) 158.47/58.09
FROM(mark(z0)) → c8(FROM(z0)) 158.47/58.09
FROM(ok(z0)) → c9(FROM(z0)) 158.47/58.09
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.47/58.09
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.47/58.09
S(mark(z0)) → c12(S(z0)) 158.47/58.09
S(ok(z0)) → c13(S(z0)) 158.47/58.09
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.47/58.09
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.47/58.09
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1)) 158.47/58.09
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0)) 158.47/58.09
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 158.47/58.09
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 158.47/58.09
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 158.47/58.09
ACTIVE(from(z0)) → c(S(z0)) 158.47/58.09
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.47/58.09
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0))) 158.47/58.09
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.47/58.09
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 158.47/58.09
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.47/58.09
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 158.47/58.09
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.47/58.09
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.47/58.09
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 158.47/58.09
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0))) 158.47/58.09
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2)))) 158.47/58.09
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 158.47/58.09
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 158.47/58.09
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 158.47/58.09
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1))) 158.47/58.09
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1))) 158.47/58.09
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.47/58.09
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0))) 158.47/58.09
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.47/58.09
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 158.47/58.09
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.47/58.09
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 158.47/58.09
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.10
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.85/58.10
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 158.85/58.10
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.10
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 158.85/58.10
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.10
PROPER(from(0)) → c17(FROM(ok(0))) 158.85/58.10
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 158.85/58.10
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 158.85/58.10
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 158.85/58.10
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1))) 158.85/58.10
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 158.85/58.10
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 158.85/58.10
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 158.85/58.10
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 158.85/58.10
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1)) 158.85/58.10
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, after, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, AFTER, 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

158.85/58.10
158.85/58.10

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

Removed 2 trailing tuple parts
158.85/58.10
158.85/58.10

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 158.85/58.10
active(after(0, z0)) → mark(z0) 158.85/58.10
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2)) 158.85/58.10
active(from(z0)) → from(active(z0)) 158.85/58.10
active(cons(z0, z1)) → cons(active(z0), z1) 158.85/58.10
active(s(z0)) → s(active(z0)) 158.85/58.10
active(after(z0, z1)) → after(active(z0), z1) 158.85/58.10
active(after(z0, z1)) → after(z0, active(z1)) 158.85/58.10
from(mark(z0)) → mark(from(z0)) 158.85/58.10
from(ok(z0)) → ok(from(z0)) 158.85/58.10
cons(mark(z0), z1) → mark(cons(z0, z1)) 158.85/58.10
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 158.85/58.10
s(mark(z0)) → mark(s(z0)) 158.85/58.10
s(ok(z0)) → ok(s(z0)) 158.85/58.10
after(mark(z0), z1) → mark(after(z0, z1)) 158.85/58.10
after(z0, mark(z1)) → mark(after(z0, z1)) 158.85/58.10
after(ok(z0), ok(z1)) → ok(after(z0, z1)) 158.85/58.10
proper(from(z0)) → from(proper(z0)) 158.85/58.10
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 158.85/58.10
proper(s(z0)) → s(proper(z0)) 158.85/58.10
proper(after(z0, z1)) → after(proper(z0), proper(z1)) 158.85/58.10
proper(0) → ok(0) 158.85/58.10
top(mark(z0)) → top(proper(z0)) 158.85/58.10
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2)) 158.85/58.10
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0)) 158.85/58.10
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1)) 158.85/58.10
FROM(mark(z0)) → c8(FROM(z0)) 158.85/58.10
FROM(ok(z0)) → c9(FROM(z0)) 158.85/58.10
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.85/58.10
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.85/58.10
S(mark(z0)) → c12(S(z0)) 158.85/58.10
S(ok(z0)) → c13(S(z0)) 158.85/58.10
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.85/58.10
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.85/58.10
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1)) 158.85/58.10
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0)) 158.85/58.10
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 158.85/58.10
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 158.85/58.10
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 158.85/58.10
ACTIVE(from(z0)) → c(S(z0)) 158.85/58.10
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.85/58.10
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.10
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.10
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 158.85/58.10
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.10
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 158.85/58.10
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.10
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.85/58.10
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 158.85/58.10
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0))) 158.85/58.10
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.10
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 158.85/58.10
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 158.85/58.10
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 158.85/58.10
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1))) 158.85/58.10
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1))) 158.85/58.10
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.85/58.10
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.10
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.10
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 158.85/58.10
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.10
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 158.85/58.10
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.10
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.85/58.10
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 158.85/58.10
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.10
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 158.85/58.10
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.10
PROPER(from(0)) → c17(FROM(ok(0))) 158.85/58.10
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 158.85/58.10
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 158.85/58.10
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 158.85/58.10
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1))) 158.85/58.10
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 158.85/58.10
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 158.85/58.10
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 158.85/58.10
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1)) 158.85/58.10
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0)) 158.85/58.10
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
S tuples:

ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2)) 158.85/58.10
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0)) 158.85/58.10
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1)) 158.85/58.10
FROM(mark(z0)) → c8(FROM(z0)) 158.85/58.10
FROM(ok(z0)) → c9(FROM(z0)) 158.85/58.10
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.85/58.10
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.85/58.10
S(mark(z0)) → c12(S(z0)) 158.85/58.10
S(ok(z0)) → c13(S(z0)) 158.85/58.10
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.85/58.10
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.85/58.10
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1)) 158.85/58.10
PROPER(s(z0)) → c19(S(proper(z0)), PROPER(z0)) 158.85/58.10
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 158.85/58.10
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 158.85/58.10
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 158.85/58.10
ACTIVE(from(z0)) → c(S(z0)) 158.85/58.10
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.85/58.10
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.10
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.10
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 158.85/58.10
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.10
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 158.85/58.10
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.10
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.85/58.10
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 158.85/58.10
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0))) 158.85/58.10
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.10
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 158.85/58.10
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 158.85/58.10
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 158.85/58.10
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1))) 158.85/58.10
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1))) 158.85/58.10
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.85/58.10
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.10
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.10
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 158.85/58.10
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.10
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 158.85/58.10
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.10
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.85/58.10
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 158.85/58.10
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.10
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 158.85/58.10
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.10
PROPER(from(0)) → c17(FROM(ok(0))) 158.85/58.10
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 158.85/58.10
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 158.85/58.10
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 158.85/58.10
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1))) 158.85/58.10
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 158.85/58.10
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 158.85/58.10
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 158.85/58.10
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1)) 158.85/58.10
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0)) 158.85/58.10
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, after, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, AFTER, 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

158.85/58.10
158.85/58.10

(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))) 158.85/58.10
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.10
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0))) 158.85/58.10
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.10
PROPER(s(0)) → c19(S(ok(0)), PROPER(0))
158.85/58.10
158.85/58.10

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 158.85/58.10
active(after(0, z0)) → mark(z0) 158.85/58.10
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2)) 158.85/58.10
active(from(z0)) → from(active(z0)) 158.85/58.10
active(cons(z0, z1)) → cons(active(z0), z1) 158.85/58.10
active(s(z0)) → s(active(z0)) 158.85/58.10
active(after(z0, z1)) → after(active(z0), z1) 158.85/58.10
active(after(z0, z1)) → after(z0, active(z1)) 158.85/58.10
from(mark(z0)) → mark(from(z0)) 158.85/58.10
from(ok(z0)) → ok(from(z0)) 158.85/58.10
cons(mark(z0), z1) → mark(cons(z0, z1)) 158.85/58.10
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 158.85/58.10
s(mark(z0)) → mark(s(z0)) 158.85/58.10
s(ok(z0)) → ok(s(z0)) 158.85/58.10
after(mark(z0), z1) → mark(after(z0, z1)) 158.85/58.10
after(z0, mark(z1)) → mark(after(z0, z1)) 158.85/58.10
after(ok(z0), ok(z1)) → ok(after(z0, z1)) 158.85/58.10
proper(from(z0)) → from(proper(z0)) 158.85/58.10
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 158.85/58.10
proper(s(z0)) → s(proper(z0)) 158.85/58.10
proper(after(z0, z1)) → after(proper(z0), proper(z1)) 158.85/58.10
proper(0) → ok(0) 158.85/58.10
top(mark(z0)) → top(proper(z0)) 158.85/58.10
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2)) 158.85/58.10
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0)) 158.85/58.10
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1)) 158.85/58.10
FROM(mark(z0)) → c8(FROM(z0)) 158.85/58.10
FROM(ok(z0)) → c9(FROM(z0)) 158.85/58.10
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.85/58.10
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.85/58.10
S(mark(z0)) → c12(S(z0)) 158.85/58.10
S(ok(z0)) → c13(S(z0)) 158.85/58.10
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.85/58.10
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.85/58.10
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1)) 158.85/58.10
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 158.85/58.10
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 158.85/58.10
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 158.85/58.10
ACTIVE(from(z0)) → c(S(z0)) 158.85/58.10
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.85/58.10
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.10
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.10
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 158.85/58.10
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.10
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 158.85/58.10
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.10
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.85/58.10
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 158.85/58.10
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0))) 158.85/58.10
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.10
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 158.85/58.10
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 158.85/58.10
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 158.85/58.10
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1))) 158.85/58.10
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1))) 158.85/58.10
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.85/58.10
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.10
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.10
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 158.85/58.10
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.10
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 158.85/58.10
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.10
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.85/58.10
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 158.85/58.10
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.10
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 158.85/58.10
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.10
PROPER(from(0)) → c17(FROM(ok(0))) 158.85/58.10
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 158.85/58.10
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 158.85/58.10
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 158.85/58.10
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1))) 158.85/58.10
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 158.85/58.10
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 158.85/58.10
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 158.85/58.10
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1)) 158.85/58.10
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0)) 158.85/58.10
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1)) 158.85/58.10
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0))) 158.85/58.10
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.10
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0))) 158.85/58.10
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.10
PROPER(s(0)) → c19(S(ok(0)), PROPER(0))
S tuples:

ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2)) 158.85/58.10
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0)) 158.85/58.10
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1)) 158.85/58.10
FROM(mark(z0)) → c8(FROM(z0)) 158.85/58.10
FROM(ok(z0)) → c9(FROM(z0)) 158.85/58.10
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.85/58.10
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.85/58.10
S(mark(z0)) → c12(S(z0)) 158.85/58.10
S(ok(z0)) → c13(S(z0)) 158.85/58.10
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.85/58.10
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.85/58.10
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1)) 158.85/58.10
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 158.85/58.10
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 158.85/58.10
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 158.85/58.10
ACTIVE(from(z0)) → c(S(z0)) 158.85/58.10
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.85/58.10
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.10
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.10
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 158.85/58.10
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.10
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 158.85/58.10
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.10
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.85/58.10
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 158.85/58.10
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0))) 158.85/58.10
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.10
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 158.85/58.10
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 158.85/58.10
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 158.85/58.10
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1))) 158.85/58.10
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1))) 158.85/58.10
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.85/58.10
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.10
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.10
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 158.85/58.10
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.10
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 158.85/58.10
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.10
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.85/58.10
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 158.85/58.10
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.10
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 158.85/58.10
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.10
PROPER(from(0)) → c17(FROM(ok(0))) 158.85/58.10
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 158.85/58.10
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 158.85/58.10
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 158.85/58.11
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1))) 158.85/58.11
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 158.85/58.11
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 158.85/58.11
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 158.85/58.11
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1)) 158.85/58.11
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0)) 158.85/58.11
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1)) 158.85/58.11
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0))) 158.85/58.11
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.11
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0))) 158.85/58.11
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.11
PROPER(s(0)) → c19(S(ok(0)), PROPER(0))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, after, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, AFTER, 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

158.85/58.11
158.85/58.11

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

Removed 1 trailing tuple parts
158.85/58.11
158.85/58.11

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 158.85/58.11
active(after(0, z0)) → mark(z0) 158.85/58.11
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2)) 158.85/58.11
active(from(z0)) → from(active(z0)) 158.85/58.11
active(cons(z0, z1)) → cons(active(z0), z1) 158.85/58.11
active(s(z0)) → s(active(z0)) 158.85/58.11
active(after(z0, z1)) → after(active(z0), z1) 158.85/58.11
active(after(z0, z1)) → after(z0, active(z1)) 158.85/58.11
from(mark(z0)) → mark(from(z0)) 158.85/58.11
from(ok(z0)) → ok(from(z0)) 158.85/58.11
cons(mark(z0), z1) → mark(cons(z0, z1)) 158.85/58.11
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 158.85/58.11
s(mark(z0)) → mark(s(z0)) 158.85/58.11
s(ok(z0)) → ok(s(z0)) 158.85/58.11
after(mark(z0), z1) → mark(after(z0, z1)) 158.85/58.11
after(z0, mark(z1)) → mark(after(z0, z1)) 158.85/58.11
after(ok(z0), ok(z1)) → ok(after(z0, z1)) 158.85/58.11
proper(from(z0)) → from(proper(z0)) 158.85/58.11
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 158.85/58.11
proper(s(z0)) → s(proper(z0)) 158.85/58.11
proper(after(z0, z1)) → after(proper(z0), proper(z1)) 158.85/58.11
proper(0) → ok(0) 158.85/58.11
top(mark(z0)) → top(proper(z0)) 158.85/58.11
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2)) 158.85/58.11
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0)) 158.85/58.11
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1)) 158.85/58.11
FROM(mark(z0)) → c8(FROM(z0)) 158.85/58.11
FROM(ok(z0)) → c9(FROM(z0)) 158.85/58.11
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.85/58.11
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.85/58.11
S(mark(z0)) → c12(S(z0)) 158.85/58.11
S(ok(z0)) → c13(S(z0)) 158.85/58.11
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.85/58.11
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.85/58.11
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1)) 158.85/58.11
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 158.85/58.11
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 158.85/58.11
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 158.85/58.11
ACTIVE(from(z0)) → c(S(z0)) 158.85/58.11
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.85/58.11
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.11
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.11
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 158.85/58.11
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.11
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 158.85/58.11
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.11
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.85/58.11
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 158.85/58.11
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0))) 158.85/58.11
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.11
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 158.85/58.11
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 158.85/58.11
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 158.85/58.11
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1))) 158.85/58.11
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1))) 158.85/58.11
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.85/58.11
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.11
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.11
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 158.85/58.11
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.11
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 158.85/58.11
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.11
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.85/58.11
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 158.85/58.11
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.11
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 158.85/58.11
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.11
PROPER(from(0)) → c17(FROM(ok(0))) 158.85/58.11
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 158.85/58.11
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 158.85/58.11
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 158.85/58.11
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1))) 158.85/58.11
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 158.85/58.11
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 158.85/58.11
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 158.85/58.11
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1)) 158.85/58.11
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0)) 158.85/58.11
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1)) 158.85/58.11
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0))) 158.85/58.11
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.11
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0))) 158.85/58.11
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.11
PROPER(s(0)) → c19(S(ok(0)))
S tuples:

ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2)) 158.85/58.11
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0)) 158.85/58.11
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1)) 158.85/58.11
FROM(mark(z0)) → c8(FROM(z0)) 158.85/58.11
FROM(ok(z0)) → c9(FROM(z0)) 158.85/58.11
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.85/58.11
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.85/58.11
S(mark(z0)) → c12(S(z0)) 158.85/58.11
S(ok(z0)) → c13(S(z0)) 158.85/58.11
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.85/58.11
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.85/58.11
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1)) 158.85/58.11
PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 158.85/58.11
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 158.85/58.11
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 158.85/58.11
ACTIVE(from(z0)) → c(S(z0)) 158.85/58.11
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.85/58.11
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.11
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.11
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 158.85/58.11
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.11
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 158.85/58.11
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.11
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.85/58.11
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 158.85/58.11
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0))) 158.85/58.11
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.11
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 158.85/58.11
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 158.85/58.11
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 158.85/58.11
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1))) 158.85/58.11
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1))) 158.85/58.11
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.85/58.11
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.11
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.11
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 158.85/58.11
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.11
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 158.85/58.11
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.11
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.85/58.11
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 158.85/58.11
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.11
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 158.85/58.11
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.11
PROPER(from(0)) → c17(FROM(ok(0))) 158.85/58.11
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 158.85/58.11
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 158.85/58.11
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 158.85/58.11
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1))) 158.85/58.12
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 158.85/58.12
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 158.85/58.12
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 158.85/58.12
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1)) 158.85/58.12
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0)) 158.85/58.12
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1)) 158.85/58.12
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0))) 158.85/58.12
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.12
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0))) 158.85/58.12
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.12
PROPER(s(0)) → c19(S(ok(0)))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, after, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, AFTER, 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

158.85/58.12
158.85/58.12

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

Use narrowing to replace PROPER(after(z0, z1)) → c20(AFTER(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(after(x0, from(z0))) → c20(AFTER(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 158.85/58.12
PROPER(after(x0, cons(z0, z1))) → c20(AFTER(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 158.85/58.12
PROPER(after(x0, s(z0))) → c20(AFTER(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 158.85/58.12
PROPER(after(x0, after(z0, z1))) → c20(AFTER(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1))) 158.85/58.12
PROPER(after(x0, 0)) → c20(AFTER(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 158.85/58.12
PROPER(after(from(z0), x1)) → c20(AFTER(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 158.85/58.12
PROPER(after(cons(z0, z1), x1)) → c20(AFTER(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 158.85/58.12
PROPER(after(s(z0), x1)) → c20(AFTER(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 158.85/58.12
PROPER(after(after(z0, z1), x1)) → c20(AFTER(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1)) 158.85/58.12
PROPER(after(0, x1)) → c20(AFTER(ok(0), proper(x1)), PROPER(0), PROPER(x1))
158.85/58.12
158.85/58.12

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 158.85/58.12
active(after(0, z0)) → mark(z0) 158.85/58.12
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2)) 158.85/58.12
active(from(z0)) → from(active(z0)) 158.85/58.12
active(cons(z0, z1)) → cons(active(z0), z1) 158.85/58.12
active(s(z0)) → s(active(z0)) 158.85/58.12
active(after(z0, z1)) → after(active(z0), z1) 158.85/58.12
active(after(z0, z1)) → after(z0, active(z1)) 158.85/58.12
from(mark(z0)) → mark(from(z0)) 158.85/58.12
from(ok(z0)) → ok(from(z0)) 158.85/58.12
cons(mark(z0), z1) → mark(cons(z0, z1)) 158.85/58.12
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 158.85/58.12
s(mark(z0)) → mark(s(z0)) 158.85/58.12
s(ok(z0)) → ok(s(z0)) 158.85/58.12
after(mark(z0), z1) → mark(after(z0, z1)) 158.85/58.12
after(z0, mark(z1)) → mark(after(z0, z1)) 158.85/58.12
after(ok(z0), ok(z1)) → ok(after(z0, z1)) 158.85/58.12
proper(from(z0)) → from(proper(z0)) 158.85/58.12
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 158.85/58.12
proper(s(z0)) → s(proper(z0)) 158.85/58.12
proper(after(z0, z1)) → after(proper(z0), proper(z1)) 158.85/58.12
proper(0) → ok(0) 158.85/58.12
top(mark(z0)) → top(proper(z0)) 158.85/58.12
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2)) 158.85/58.12
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0)) 158.85/58.12
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1)) 158.85/58.12
FROM(mark(z0)) → c8(FROM(z0)) 158.85/58.12
FROM(ok(z0)) → c9(FROM(z0)) 158.85/58.12
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.85/58.12
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.85/58.12
S(mark(z0)) → c12(S(z0)) 158.85/58.12
S(ok(z0)) → c13(S(z0)) 158.85/58.12
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.85/58.12
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.85/58.12
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1)) 158.85/58.12
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 158.85/58.12
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 158.85/58.12
ACTIVE(from(z0)) → c(S(z0)) 158.85/58.12
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.85/58.12
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.12
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.12
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 158.85/58.12
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.12
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 158.85/58.12
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.12
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.85/58.12
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 158.85/58.12
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0))) 158.85/58.12
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.12
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 158.85/58.12
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 158.85/58.12
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 158.85/58.12
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1))) 158.85/58.12
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1))) 158.85/58.12
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.85/58.12
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.12
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.12
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 158.85/58.12
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.12
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 158.85/58.12
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.12
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.85/58.12
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 158.85/58.12
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.12
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 158.85/58.12
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.12
PROPER(from(0)) → c17(FROM(ok(0))) 158.85/58.12
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 158.85/58.12
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 158.85/58.12
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 158.85/58.12
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1))) 158.85/58.12
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 158.85/58.12
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 158.85/58.12
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 158.85/58.12
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1)) 158.85/58.12
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0)) 158.85/58.12
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1)) 158.85/58.12
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0))) 158.85/58.12
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.12
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0))) 158.85/58.12
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.12
PROPER(s(0)) → c19(S(ok(0))) 158.85/58.12
PROPER(after(x0, from(z0))) → c20(AFTER(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 158.85/58.12
PROPER(after(x0, cons(z0, z1))) → c20(AFTER(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 158.85/58.12
PROPER(after(x0, s(z0))) → c20(AFTER(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 158.85/58.12
PROPER(after(x0, after(z0, z1))) → c20(AFTER(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1))) 158.85/58.12
PROPER(after(x0, 0)) → c20(AFTER(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 158.85/58.12
PROPER(after(from(z0), x1)) → c20(AFTER(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 158.85/58.12
PROPER(after(cons(z0, z1), x1)) → c20(AFTER(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 158.85/58.12
PROPER(after(s(z0), x1)) → c20(AFTER(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 158.85/58.12
PROPER(after(after(z0, z1), x1)) → c20(AFTER(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1)) 158.85/58.12
PROPER(after(0, x1)) → c20(AFTER(ok(0), proper(x1)), PROPER(0), PROPER(x1))
S tuples:

ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2)) 158.85/58.12
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0)) 158.85/58.12
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1)) 158.85/58.12
FROM(mark(z0)) → c8(FROM(z0)) 158.85/58.12
FROM(ok(z0)) → c9(FROM(z0)) 158.85/58.12
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.85/58.12
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.85/58.12
S(mark(z0)) → c12(S(z0)) 158.85/58.12
S(ok(z0)) → c13(S(z0)) 158.85/58.12
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.85/58.12
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.85/58.12
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1)) 158.85/58.12
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 158.85/58.12
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 158.85/58.12
ACTIVE(from(z0)) → c(S(z0)) 158.85/58.12
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.85/58.12
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.12
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.12
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 158.85/58.12
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.12
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 158.85/58.12
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.12
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.85/58.12
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 158.85/58.12
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0))) 158.85/58.12
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.12
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 158.85/58.12
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 158.85/58.12
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 158.85/58.12
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1))) 158.85/58.12
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1))) 158.85/58.12
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.85/58.12
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.12
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.12
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 158.85/58.12
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.12
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 158.85/58.12
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.12
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.85/58.12
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 158.85/58.12
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.12
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 158.85/58.12
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.12
PROPER(from(0)) → c17(FROM(ok(0))) 158.85/58.12
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 158.85/58.12
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 158.85/58.12
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 158.85/58.12
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1))) 158.85/58.12
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 158.85/58.12
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 158.85/58.12
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 158.85/58.12
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1)) 158.85/58.12
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0)) 158.85/58.12
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1)) 158.85/58.12
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0))) 158.85/58.12
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.12
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0))) 158.85/58.12
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.12
PROPER(s(0)) → c19(S(ok(0))) 158.85/58.12
PROPER(after(x0, from(z0))) → c20(AFTER(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 158.85/58.12
PROPER(after(x0, cons(z0, z1))) → c20(AFTER(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 158.85/58.12
PROPER(after(x0, s(z0))) → c20(AFTER(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 158.85/58.12
PROPER(after(x0, after(z0, z1))) → c20(AFTER(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1))) 158.85/58.12
PROPER(after(x0, 0)) → c20(AFTER(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 158.85/58.12
PROPER(after(from(z0), x1)) → c20(AFTER(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 158.85/58.12
PROPER(after(cons(z0, z1), x1)) → c20(AFTER(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 158.85/58.12
PROPER(after(s(z0), x1)) → c20(AFTER(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 158.85/58.12
PROPER(after(after(z0, z1), x1)) → c20(AFTER(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1)) 158.85/58.12
PROPER(after(0, x1)) → c20(AFTER(ok(0), proper(x1)), PROPER(0), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, after, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, AFTER, 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

158.85/58.12
158.85/58.12

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

Removed 2 trailing tuple parts
158.85/58.12
158.85/58.12

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 158.85/58.12
active(after(0, z0)) → mark(z0) 158.85/58.12
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2)) 158.85/58.12
active(from(z0)) → from(active(z0)) 158.85/58.12
active(cons(z0, z1)) → cons(active(z0), z1) 158.85/58.12
active(s(z0)) → s(active(z0)) 158.85/58.12
active(after(z0, z1)) → after(active(z0), z1) 158.85/58.12
active(after(z0, z1)) → after(z0, active(z1)) 158.85/58.12
from(mark(z0)) → mark(from(z0)) 158.85/58.12
from(ok(z0)) → ok(from(z0)) 158.85/58.12
cons(mark(z0), z1) → mark(cons(z0, z1)) 158.85/58.12
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 158.85/58.12
s(mark(z0)) → mark(s(z0)) 158.85/58.12
s(ok(z0)) → ok(s(z0)) 158.85/58.12
after(mark(z0), z1) → mark(after(z0, z1)) 158.85/58.12
after(z0, mark(z1)) → mark(after(z0, z1)) 158.85/58.12
after(ok(z0), ok(z1)) → ok(after(z0, z1)) 158.85/58.12
proper(from(z0)) → from(proper(z0)) 158.85/58.12
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 158.85/58.12
proper(s(z0)) → s(proper(z0)) 158.85/58.12
proper(after(z0, z1)) → after(proper(z0), proper(z1)) 158.85/58.12
proper(0) → ok(0) 158.85/58.12
top(mark(z0)) → top(proper(z0)) 158.85/58.12
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2)) 158.85/58.12
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0)) 158.85/58.12
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1)) 158.85/58.12
FROM(mark(z0)) → c8(FROM(z0)) 158.85/58.12
FROM(ok(z0)) → c9(FROM(z0)) 158.85/58.12
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.85/58.12
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.85/58.12
S(mark(z0)) → c12(S(z0)) 158.85/58.12
S(ok(z0)) → c13(S(z0)) 158.85/58.12
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.85/58.12
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.85/58.12
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1)) 158.85/58.12
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 158.85/58.12
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 158.85/58.12
ACTIVE(from(z0)) → c(S(z0)) 158.85/58.12
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.85/58.12
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.12
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.12
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 158.85/58.12
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.12
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 158.85/58.12
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.12
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.85/58.12
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 158.85/58.12
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0))) 158.85/58.12
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.12
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 158.85/58.12
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 158.85/58.12
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 158.85/58.12
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1))) 158.85/58.12
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1))) 158.85/58.12
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.85/58.12
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.12
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.12
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 158.85/58.12
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.12
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 158.85/58.12
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.12
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.85/58.12
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 158.85/58.12
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.12
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 158.85/58.12
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.12
PROPER(from(0)) → c17(FROM(ok(0))) 158.85/58.12
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 158.85/58.12
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 158.85/58.12
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 158.85/58.12
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1))) 158.85/58.12
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 158.85/58.12
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 158.85/58.12
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 158.85/58.12
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1)) 158.85/58.12
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0)) 158.85/58.12
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1)) 158.85/58.12
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0))) 158.85/58.12
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.12
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0))) 158.85/58.12
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.12
PROPER(s(0)) → c19(S(ok(0))) 158.85/58.12
PROPER(after(x0, from(z0))) → c20(AFTER(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 158.85/58.12
PROPER(after(x0, cons(z0, z1))) → c20(AFTER(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 158.85/58.12
PROPER(after(x0, s(z0))) → c20(AFTER(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 158.85/58.12
PROPER(after(x0, after(z0, z1))) → c20(AFTER(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1))) 158.85/58.12
PROPER(after(from(z0), x1)) → c20(AFTER(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 158.85/58.12
PROPER(after(cons(z0, z1), x1)) → c20(AFTER(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 158.85/58.12
PROPER(after(s(z0), x1)) → c20(AFTER(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 158.85/58.12
PROPER(after(after(z0, z1), x1)) → c20(AFTER(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1)) 158.85/58.12
PROPER(after(x0, 0)) → c20(AFTER(proper(x0), ok(0)), PROPER(x0)) 158.85/58.12
PROPER(after(0, x1)) → c20(AFTER(ok(0), proper(x1)), PROPER(x1))
S tuples:

ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2)) 158.85/58.12
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0)) 158.85/58.12
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1)) 158.85/58.12
FROM(mark(z0)) → c8(FROM(z0)) 158.85/58.12
FROM(ok(z0)) → c9(FROM(z0)) 158.85/58.12
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.85/58.12
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.85/58.12
S(mark(z0)) → c12(S(z0)) 158.85/58.12
S(ok(z0)) → c13(S(z0)) 158.85/58.12
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.85/58.12
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.85/58.12
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1)) 158.85/58.12
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0)) 158.85/58.12
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 158.85/58.12
ACTIVE(from(z0)) → c(S(z0)) 158.85/58.12
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.85/58.12
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.12
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.12
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 158.85/58.12
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.12
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 158.85/58.12
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.12
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.85/58.12
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 158.85/58.12
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0))) 158.85/58.12
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.12
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 158.85/58.12
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 158.85/58.12
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 158.85/58.12
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1))) 158.85/58.12
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1))) 158.85/58.12
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.85/58.12
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.12
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.12
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 158.85/58.12
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.12
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 158.85/58.12
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.12
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.85/58.13
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 158.85/58.13
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.13
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 158.85/58.13
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.13
PROPER(from(0)) → c17(FROM(ok(0))) 158.85/58.13
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 158.85/58.13
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 158.85/58.13
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 158.85/58.13
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1))) 158.85/58.13
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 158.85/58.13
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 158.85/58.13
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 158.85/58.13
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1)) 158.85/58.13
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0)) 158.85/58.13
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1)) 158.85/58.13
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0))) 158.85/58.13
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.13
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0))) 158.85/58.13
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.13
PROPER(s(0)) → c19(S(ok(0))) 158.85/58.13
PROPER(after(x0, from(z0))) → c20(AFTER(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 158.85/58.13
PROPER(after(x0, cons(z0, z1))) → c20(AFTER(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 158.85/58.13
PROPER(after(x0, s(z0))) → c20(AFTER(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 158.85/58.13
PROPER(after(x0, after(z0, z1))) → c20(AFTER(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1))) 158.85/58.13
PROPER(after(from(z0), x1)) → c20(AFTER(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 158.85/58.13
PROPER(after(cons(z0, z1), x1)) → c20(AFTER(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 158.85/58.13
PROPER(after(s(z0), x1)) → c20(AFTER(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 158.85/58.13
PROPER(after(after(z0, z1), x1)) → c20(AFTER(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1)) 158.85/58.13
PROPER(after(x0, 0)) → c20(AFTER(proper(x0), ok(0)), PROPER(x0)) 158.85/58.13
PROPER(after(0, x1)) → c20(AFTER(ok(0), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, after, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, AFTER, 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

158.85/58.13
158.85/58.13

(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))) 158.85/58.13
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.13
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0))) 158.85/58.13
TOP(mark(after(z0, z1))) → c22(TOP(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.13
TOP(mark(0)) → c22(TOP(ok(0)), PROPER(0))
158.85/58.13
158.85/58.13

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 158.85/58.13
active(after(0, z0)) → mark(z0) 158.85/58.13
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2)) 158.85/58.13
active(from(z0)) → from(active(z0)) 158.85/58.13
active(cons(z0, z1)) → cons(active(z0), z1) 158.85/58.13
active(s(z0)) → s(active(z0)) 158.85/58.13
active(after(z0, z1)) → after(active(z0), z1) 158.85/58.13
active(after(z0, z1)) → after(z0, active(z1)) 158.85/58.13
from(mark(z0)) → mark(from(z0)) 158.85/58.13
from(ok(z0)) → ok(from(z0)) 158.85/58.13
cons(mark(z0), z1) → mark(cons(z0, z1)) 158.85/58.13
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 158.85/58.13
s(mark(z0)) → mark(s(z0)) 158.85/58.13
s(ok(z0)) → ok(s(z0)) 158.85/58.13
after(mark(z0), z1) → mark(after(z0, z1)) 158.85/58.13
after(z0, mark(z1)) → mark(after(z0, z1)) 158.85/58.13
after(ok(z0), ok(z1)) → ok(after(z0, z1)) 158.85/58.13
proper(from(z0)) → from(proper(z0)) 158.85/58.13
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 158.85/58.13
proper(s(z0)) → s(proper(z0)) 158.85/58.13
proper(after(z0, z1)) → after(proper(z0), proper(z1)) 158.85/58.13
proper(0) → ok(0) 158.85/58.13
top(mark(z0)) → top(proper(z0)) 158.85/58.13
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2)) 158.85/58.13
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0)) 158.85/58.13
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1)) 158.85/58.13
FROM(mark(z0)) → c8(FROM(z0)) 158.85/58.13
FROM(ok(z0)) → c9(FROM(z0)) 158.85/58.13
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.85/58.13
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.85/58.13
S(mark(z0)) → c12(S(z0)) 158.85/58.13
S(ok(z0)) → c13(S(z0)) 158.85/58.13
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.85/58.13
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.85/58.13
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1)) 158.85/58.13
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 158.85/58.13
ACTIVE(from(z0)) → c(S(z0)) 158.85/58.13
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.85/58.13
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.13
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.13
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 158.85/58.13
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.13
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 158.85/58.13
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.13
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.85/58.13
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 158.85/58.13
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0))) 158.85/58.13
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.13
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 158.85/58.13
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 158.85/58.13
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 158.85/58.13
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1))) 158.85/58.13
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1))) 158.85/58.13
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.85/58.13
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.13
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.13
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 158.85/58.13
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.13
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 158.85/58.13
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.13
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.85/58.13
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 158.85/58.13
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.13
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 158.85/58.13
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.13
PROPER(from(0)) → c17(FROM(ok(0))) 158.85/58.13
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 158.85/58.13
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 158.85/58.13
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 158.85/58.13
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1))) 158.85/58.13
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 158.85/58.13
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 158.85/58.13
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 158.85/58.13
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1)) 158.85/58.13
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0)) 158.85/58.13
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1)) 158.85/58.13
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0))) 158.85/58.13
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.13
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0))) 158.85/58.13
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.13
PROPER(s(0)) → c19(S(ok(0))) 158.85/58.13
PROPER(after(x0, from(z0))) → c20(AFTER(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 158.85/58.13
PROPER(after(x0, cons(z0, z1))) → c20(AFTER(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 158.85/58.13
PROPER(after(x0, s(z0))) → c20(AFTER(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 158.85/58.13
PROPER(after(x0, after(z0, z1))) → c20(AFTER(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1))) 158.85/58.13
PROPER(after(from(z0), x1)) → c20(AFTER(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 158.85/58.13
PROPER(after(cons(z0, z1), x1)) → c20(AFTER(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 158.85/58.13
PROPER(after(s(z0), x1)) → c20(AFTER(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 158.85/58.13
PROPER(after(after(z0, z1), x1)) → c20(AFTER(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1)) 158.85/58.13
PROPER(after(x0, 0)) → c20(AFTER(proper(x0), ok(0)), PROPER(x0)) 158.85/58.13
PROPER(after(0, x1)) → c20(AFTER(ok(0), proper(x1)), PROPER(x1)) 158.85/58.13
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0))) 158.85/58.13
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.13
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0))) 158.85/58.13
TOP(mark(after(z0, z1))) → c22(TOP(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.13
TOP(mark(0)) → c22(TOP(ok(0)), PROPER(0))
S tuples:

ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2)) 158.85/58.13
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0)) 158.85/58.13
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1)) 158.85/58.13
FROM(mark(z0)) → c8(FROM(z0)) 158.85/58.13
FROM(ok(z0)) → c9(FROM(z0)) 158.85/58.13
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.85/58.13
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.85/58.13
S(mark(z0)) → c12(S(z0)) 158.85/58.13
S(ok(z0)) → c13(S(z0)) 158.85/58.13
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.85/58.13
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.85/58.13
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1)) 158.85/58.13
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 158.85/58.13
ACTIVE(from(z0)) → c(S(z0)) 158.85/58.13
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.85/58.13
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.13
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.13
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 158.85/58.13
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.13
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 158.85/58.13
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.13
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.85/58.13
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 158.85/58.13
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0))) 158.85/58.13
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.13
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 158.85/58.13
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 158.85/58.13
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 158.85/58.13
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1))) 158.85/58.13
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1))) 158.85/58.13
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.85/58.13
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.13
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.13
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 158.85/58.13
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.13
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 158.85/58.13
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.13
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.85/58.13
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 158.85/58.13
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.13
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 158.85/58.13
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.13
PROPER(from(0)) → c17(FROM(ok(0))) 158.85/58.13
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 158.85/58.13
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 158.85/58.13
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 158.85/58.13
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1))) 158.85/58.13
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 158.85/58.13
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 158.85/58.13
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 158.85/58.13
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1)) 158.85/58.13
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0)) 158.85/58.13
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1)) 158.85/58.13
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0))) 158.85/58.15
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.15
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0))) 158.85/58.15
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.15
PROPER(s(0)) → c19(S(ok(0))) 158.85/58.15
PROPER(after(x0, from(z0))) → c20(AFTER(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 158.85/58.15
PROPER(after(x0, cons(z0, z1))) → c20(AFTER(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 158.85/58.15
PROPER(after(x0, s(z0))) → c20(AFTER(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 158.85/58.15
PROPER(after(x0, after(z0, z1))) → c20(AFTER(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1))) 158.85/58.15
PROPER(after(from(z0), x1)) → c20(AFTER(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 158.85/58.15
PROPER(after(cons(z0, z1), x1)) → c20(AFTER(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 158.85/58.15
PROPER(after(s(z0), x1)) → c20(AFTER(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 158.85/58.15
PROPER(after(after(z0, z1), x1)) → c20(AFTER(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1)) 158.85/58.15
PROPER(after(x0, 0)) → c20(AFTER(proper(x0), ok(0)), PROPER(x0)) 158.85/58.15
PROPER(after(0, x1)) → c20(AFTER(ok(0), proper(x1)), PROPER(x1)) 158.85/58.15
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0))) 158.85/58.15
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.15
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0))) 158.85/58.15
TOP(mark(after(z0, z1))) → c22(TOP(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.15
TOP(mark(0)) → c22(TOP(ok(0)), PROPER(0))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, after, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, AFTER, 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

158.85/58.15
158.85/58.15

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

Removed 1 trailing tuple parts
158.85/58.15
158.85/58.15

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 158.85/58.15
active(after(0, z0)) → mark(z0) 158.85/58.15
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2)) 158.85/58.15
active(from(z0)) → from(active(z0)) 158.85/58.15
active(cons(z0, z1)) → cons(active(z0), z1) 158.85/58.15
active(s(z0)) → s(active(z0)) 158.85/58.15
active(after(z0, z1)) → after(active(z0), z1) 158.85/58.15
active(after(z0, z1)) → after(z0, active(z1)) 158.85/58.15
from(mark(z0)) → mark(from(z0)) 158.85/58.15
from(ok(z0)) → ok(from(z0)) 158.85/58.15
cons(mark(z0), z1) → mark(cons(z0, z1)) 158.85/58.15
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 158.85/58.15
s(mark(z0)) → mark(s(z0)) 158.85/58.15
s(ok(z0)) → ok(s(z0)) 158.85/58.15
after(mark(z0), z1) → mark(after(z0, z1)) 158.85/58.15
after(z0, mark(z1)) → mark(after(z0, z1)) 158.85/58.15
after(ok(z0), ok(z1)) → ok(after(z0, z1)) 158.85/58.15
proper(from(z0)) → from(proper(z0)) 158.85/58.15
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 158.85/58.15
proper(s(z0)) → s(proper(z0)) 158.85/58.15
proper(after(z0, z1)) → after(proper(z0), proper(z1)) 158.85/58.15
proper(0) → ok(0) 158.85/58.15
top(mark(z0)) → top(proper(z0)) 158.85/58.15
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2)) 158.85/58.15
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0)) 158.85/58.15
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1)) 158.85/58.15
FROM(mark(z0)) → c8(FROM(z0)) 158.85/58.15
FROM(ok(z0)) → c9(FROM(z0)) 158.85/58.15
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.85/58.15
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.85/58.15
S(mark(z0)) → c12(S(z0)) 158.85/58.15
S(ok(z0)) → c13(S(z0)) 158.85/58.15
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.85/58.15
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.85/58.15
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1)) 158.85/58.15
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 158.85/58.15
ACTIVE(from(z0)) → c(S(z0)) 158.85/58.15
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.85/58.15
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.15
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.15
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 158.85/58.15
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.15
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 158.85/58.15
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.15
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.85/58.15
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 158.85/58.15
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0))) 158.85/58.15
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.15
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 158.85/58.15
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 158.85/58.15
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 158.85/58.15
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1))) 158.85/58.15
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1))) 158.85/58.15
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.85/58.15
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.15
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.15
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 158.85/58.15
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.15
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 158.85/58.15
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.15
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.85/58.15
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 158.85/58.15
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.15
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 158.85/58.15
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.15
PROPER(from(0)) → c17(FROM(ok(0))) 158.85/58.15
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 158.85/58.15
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 158.85/58.15
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 158.85/58.15
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1))) 158.85/58.15
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 158.85/58.15
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 158.85/58.15
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 158.85/58.15
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1)) 158.85/58.15
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0)) 158.85/58.15
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1)) 158.85/58.15
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0))) 158.85/58.15
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.15
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0))) 158.85/58.15
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.15
PROPER(s(0)) → c19(S(ok(0))) 158.85/58.15
PROPER(after(x0, from(z0))) → c20(AFTER(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 158.85/58.15
PROPER(after(x0, cons(z0, z1))) → c20(AFTER(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 158.85/58.15
PROPER(after(x0, s(z0))) → c20(AFTER(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 158.85/58.15
PROPER(after(x0, after(z0, z1))) → c20(AFTER(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1))) 158.85/58.15
PROPER(after(from(z0), x1)) → c20(AFTER(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 158.85/58.15
PROPER(after(cons(z0, z1), x1)) → c20(AFTER(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 158.85/58.15
PROPER(after(s(z0), x1)) → c20(AFTER(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 158.85/58.15
PROPER(after(after(z0, z1), x1)) → c20(AFTER(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1)) 158.85/58.15
PROPER(after(x0, 0)) → c20(AFTER(proper(x0), ok(0)), PROPER(x0)) 158.85/58.15
PROPER(after(0, x1)) → c20(AFTER(ok(0), proper(x1)), PROPER(x1)) 158.85/58.15
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0))) 158.85/58.15
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.15
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0))) 158.85/58.15
TOP(mark(after(z0, z1))) → c22(TOP(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.15
TOP(mark(0)) → c22(TOP(ok(0)))
S tuples:

ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2)) 158.85/58.15
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0)) 158.85/58.15
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1)) 158.85/58.15
FROM(mark(z0)) → c8(FROM(z0)) 158.85/58.15
FROM(ok(z0)) → c9(FROM(z0)) 158.85/58.15
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.85/58.15
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.85/58.15
S(mark(z0)) → c12(S(z0)) 158.85/58.15
S(ok(z0)) → c13(S(z0)) 158.85/58.15
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.85/58.15
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.85/58.15
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1)) 158.85/58.15
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 158.85/58.15
ACTIVE(from(z0)) → c(S(z0)) 158.85/58.15
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.85/58.15
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.15
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.15
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 158.85/58.15
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.15
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 158.85/58.15
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.15
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.85/58.15
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 158.85/58.15
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0))) 158.85/58.15
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.15
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 158.85/58.15
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 158.85/58.16
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 158.85/58.16
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1))) 158.85/58.16
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1))) 158.85/58.16
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.85/58.16
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.16
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.16
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 158.85/58.16
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.16
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 158.85/58.16
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.16
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.85/58.16
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 158.85/58.16
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.16
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 158.85/58.16
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.16
PROPER(from(0)) → c17(FROM(ok(0))) 158.85/58.16
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 158.85/58.16
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 158.85/58.16
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 158.85/58.16
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1))) 158.85/58.16
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 158.85/58.16
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 158.85/58.16
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 158.85/58.16
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1)) 158.85/58.16
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0)) 158.85/58.16
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1)) 158.85/58.16
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0))) 158.85/58.16
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.16
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0))) 158.85/58.16
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.16
PROPER(s(0)) → c19(S(ok(0))) 158.85/58.16
PROPER(after(x0, from(z0))) → c20(AFTER(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 158.85/58.16
PROPER(after(x0, cons(z0, z1))) → c20(AFTER(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 158.85/58.16
PROPER(after(x0, s(z0))) → c20(AFTER(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 158.85/58.16
PROPER(after(x0, after(z0, z1))) → c20(AFTER(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1))) 158.85/58.16
PROPER(after(from(z0), x1)) → c20(AFTER(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 158.85/58.16
PROPER(after(cons(z0, z1), x1)) → c20(AFTER(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 158.85/58.16
PROPER(after(s(z0), x1)) → c20(AFTER(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 158.85/58.16
PROPER(after(after(z0, z1), x1)) → c20(AFTER(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1)) 158.85/58.16
PROPER(after(x0, 0)) → c20(AFTER(proper(x0), ok(0)), PROPER(x0)) 158.85/58.16
PROPER(after(0, x1)) → c20(AFTER(ok(0), proper(x1)), PROPER(x1)) 158.85/58.16
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0))) 158.85/58.16
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.16
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0))) 158.85/58.16
TOP(mark(after(z0, z1))) → c22(TOP(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.16
TOP(mark(0)) → c22(TOP(ok(0)))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, after, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, AFTER, 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

158.85/58.16
158.85/58.16

(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)) 158.85/58.16
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 158.85/58.16
proper(s(z0)) → s(proper(z0)) 158.85/58.16
proper(after(z0, z1)) → after(proper(z0), proper(z1)) 158.85/58.16
proper(0) → ok(0) 158.85/58.16
after(mark(z0), z1) → mark(after(z0, z1)) 158.85/58.16
after(z0, mark(z1)) → mark(after(z0, z1)) 158.85/58.16
after(ok(z0), ok(z1)) → ok(after(z0, z1)) 158.85/58.16
s(mark(z0)) → mark(s(z0)) 158.85/58.16
s(ok(z0)) → ok(s(z0)) 158.85/58.16
cons(mark(z0), z1) → mark(cons(z0, z1)) 158.85/58.16
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 158.85/58.16
from(mark(z0)) → mark(from(z0)) 158.85/58.16
from(ok(z0)) → ok(from(z0)) 158.85/58.16
active(from(z0)) → mark(cons(z0, from(s(z0)))) 158.85/58.16
active(after(0, z0)) → mark(z0) 158.85/58.16
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2)) 158.85/58.16
active(from(z0)) → from(active(z0)) 158.85/58.16
active(cons(z0, z1)) → cons(active(z0), z1) 158.85/58.16
active(s(z0)) → s(active(z0)) 158.85/58.16
active(after(z0, z1)) → after(active(z0), z1) 158.85/58.16
active(after(z0, z1)) → after(z0, active(z1))
And the Tuples:

ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2)) 158.85/58.16
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0)) 158.85/58.16
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1)) 158.85/58.16
FROM(mark(z0)) → c8(FROM(z0)) 158.85/58.16
FROM(ok(z0)) → c9(FROM(z0)) 158.85/58.16
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.85/58.16
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.85/58.16
S(mark(z0)) → c12(S(z0)) 158.85/58.16
S(ok(z0)) → c13(S(z0)) 158.85/58.16
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.85/58.16
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.85/58.16
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1)) 158.85/58.16
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 158.85/58.16
ACTIVE(from(z0)) → c(S(z0)) 158.85/58.16
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.85/58.16
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.16
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.16
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 158.85/58.16
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.16
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 158.85/58.16
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.16
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.85/58.16
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 158.85/58.16
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0))) 158.85/58.16
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.16
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 158.85/58.16
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 158.85/58.16
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 158.85/58.16
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1))) 158.85/58.16
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1))) 158.85/58.16
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.85/58.16
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.16
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.16
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 158.85/58.16
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.16
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 158.85/58.16
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.16
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.85/58.16
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 158.85/58.16
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.16
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 158.85/58.16
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.16
PROPER(from(0)) → c17(FROM(ok(0))) 158.85/58.16
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 158.85/58.16
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 158.85/58.16
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 158.85/58.16
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1))) 158.85/58.16
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 158.85/58.16
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 158.85/58.16
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 158.85/58.16
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1)) 158.85/58.16
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0)) 158.85/58.16
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1)) 158.85/58.16
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0))) 158.85/58.16
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.16
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0))) 158.85/58.16
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.16
PROPER(s(0)) → c19(S(ok(0))) 158.85/58.16
PROPER(after(x0, from(z0))) → c20(AFTER(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 158.85/58.16
PROPER(after(x0, cons(z0, z1))) → c20(AFTER(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 158.85/58.16
PROPER(after(x0, s(z0))) → c20(AFTER(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 158.85/58.16
PROPER(after(x0, after(z0, z1))) → c20(AFTER(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1))) 158.85/58.16
PROPER(after(from(z0), x1)) → c20(AFTER(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 158.85/58.16
PROPER(after(cons(z0, z1), x1)) → c20(AFTER(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 158.85/58.16
PROPER(after(s(z0), x1)) → c20(AFTER(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 158.85/58.16
PROPER(after(after(z0, z1), x1)) → c20(AFTER(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1)) 158.85/58.18
PROPER(after(x0, 0)) → c20(AFTER(proper(x0), ok(0)), PROPER(x0)) 158.85/58.18
PROPER(after(0, x1)) → c20(AFTER(ok(0), proper(x1)), PROPER(x1)) 158.85/58.18
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0))) 158.85/58.18
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.18
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0))) 158.85/58.18
TOP(mark(after(z0, z1))) → c22(TOP(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.18
TOP(mark(0)) → c22(TOP(ok(0)))
The order we found is given by the following interpretation:
Polynomial interpretation : 158.85/58.18

POL(0) = 0    158.85/58.18
POL(ACTIVE(x1)) = 0    158.85/58.18
POL(AFTER(x1, x2)) = 0    158.85/58.18
POL(CONS(x1, x2)) = 0    158.85/58.18
POL(FROM(x1)) = 0    158.85/58.18
POL(PROPER(x1)) = 0    158.85/58.18
POL(S(x1)) = 0    158.85/58.18
POL(TOP(x1)) = [2]x1    158.85/58.18
POL(active(x1)) = x1    158.85/58.18
POL(after(x1, x2)) = [2]    158.85/58.18
POL(c(x1)) = x1    158.85/58.18
POL(c10(x1)) = x1    158.85/58.18
POL(c11(x1)) = x1    158.85/58.18
POL(c12(x1)) = x1    158.85/58.18
POL(c13(x1)) = x1    158.85/58.18
POL(c14(x1)) = x1    158.85/58.18
POL(c15(x1)) = x1    158.85/58.18
POL(c16(x1)) = x1    158.85/58.18
POL(c17(x1)) = x1    158.85/58.18
POL(c17(x1, x2)) = x1 + x2    158.85/58.18
POL(c18(x1, x2)) = x1 + x2    158.85/58.18
POL(c18(x1, x2, x3)) = x1 + x2 + x3    158.85/58.18
POL(c19(x1)) = x1    158.85/58.18
POL(c19(x1, x2)) = x1 + x2    158.85/58.18
POL(c2(x1)) = x1    158.85/58.18
POL(c20(x1, x2)) = x1 + x2    158.85/58.18
POL(c20(x1, x2, x3)) = x1 + x2 + x3    158.85/58.18
POL(c22(x1)) = x1    158.85/58.18
POL(c22(x1, x2)) = x1 + x2    158.85/58.18
POL(c23(x1, x2)) = x1 + x2    158.85/58.18
POL(c3(x1, x2)) = x1 + x2    158.85/58.18
POL(c4(x1, x2)) = x1 + x2    158.85/58.18
POL(c5(x1, x2)) = x1 + x2    158.85/58.18
POL(c6(x1, x2)) = x1 + x2    158.85/58.18
POL(c7(x1, x2)) = x1 + x2    158.85/58.18
POL(c8(x1)) = x1    158.85/58.18
POL(c9(x1)) = x1    158.85/58.18
POL(cons(x1, x2)) = [2]    158.85/58.18
POL(from(x1)) = [2]    158.85/58.18
POL(mark(x1)) = [2]    158.85/58.18
POL(ok(x1)) = x1    158.85/58.18
POL(proper(x1)) = 0    158.85/58.18
POL(s(x1)) = [2]   
158.85/58.18
158.85/58.18

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 158.85/58.18
active(after(0, z0)) → mark(z0) 158.85/58.18
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2)) 158.85/58.18
active(from(z0)) → from(active(z0)) 158.85/58.18
active(cons(z0, z1)) → cons(active(z0), z1) 158.85/58.18
active(s(z0)) → s(active(z0)) 158.85/58.18
active(after(z0, z1)) → after(active(z0), z1) 158.85/58.18
active(after(z0, z1)) → after(z0, active(z1)) 158.85/58.18
from(mark(z0)) → mark(from(z0)) 158.85/58.18
from(ok(z0)) → ok(from(z0)) 158.85/58.18
cons(mark(z0), z1) → mark(cons(z0, z1)) 158.85/58.18
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 158.85/58.18
s(mark(z0)) → mark(s(z0)) 158.85/58.18
s(ok(z0)) → ok(s(z0)) 158.85/58.18
after(mark(z0), z1) → mark(after(z0, z1)) 158.85/58.18
after(z0, mark(z1)) → mark(after(z0, z1)) 158.85/58.18
after(ok(z0), ok(z1)) → ok(after(z0, z1)) 158.85/58.18
proper(from(z0)) → from(proper(z0)) 158.85/58.18
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 158.85/58.18
proper(s(z0)) → s(proper(z0)) 158.85/58.18
proper(after(z0, z1)) → after(proper(z0), proper(z1)) 158.85/58.18
proper(0) → ok(0) 158.85/58.18
top(mark(z0)) → top(proper(z0)) 158.85/58.18
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2)) 158.85/58.18
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0)) 158.85/58.18
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1)) 158.85/58.18
FROM(mark(z0)) → c8(FROM(z0)) 158.85/58.18
FROM(ok(z0)) → c9(FROM(z0)) 158.85/58.18
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.85/58.18
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.85/58.18
S(mark(z0)) → c12(S(z0)) 158.85/58.18
S(ok(z0)) → c13(S(z0)) 158.85/58.18
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.85/58.18
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.85/58.18
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1)) 158.85/58.18
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 158.85/58.18
ACTIVE(from(z0)) → c(S(z0)) 158.85/58.18
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.85/58.18
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.18
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.18
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 158.85/58.18
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.18
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 158.85/58.18
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.18
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.85/58.18
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 158.85/58.18
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0))) 158.85/58.18
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.18
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 158.85/58.18
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 158.85/58.18
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 158.85/58.18
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1))) 158.85/58.18
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1))) 158.85/58.18
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.85/58.18
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.18
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.18
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 158.85/58.18
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.18
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 158.85/58.18
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.18
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.85/58.18
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 158.85/58.18
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.18
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 158.85/58.18
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.18
PROPER(from(0)) → c17(FROM(ok(0))) 158.85/58.18
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 158.85/58.18
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 158.85/58.18
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 158.85/58.18
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1))) 158.85/58.18
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 158.85/58.18
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 158.85/58.18
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 158.85/58.18
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1)) 158.85/58.18
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0)) 158.85/58.18
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1)) 158.85/58.18
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0))) 158.85/58.18
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.18
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0))) 158.85/58.18
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.18
PROPER(s(0)) → c19(S(ok(0))) 158.85/58.18
PROPER(after(x0, from(z0))) → c20(AFTER(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 158.85/58.18
PROPER(after(x0, cons(z0, z1))) → c20(AFTER(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 158.85/58.18
PROPER(after(x0, s(z0))) → c20(AFTER(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 158.85/58.18
PROPER(after(x0, after(z0, z1))) → c20(AFTER(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1))) 158.85/58.18
PROPER(after(from(z0), x1)) → c20(AFTER(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 158.85/58.18
PROPER(after(cons(z0, z1), x1)) → c20(AFTER(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 158.85/58.18
PROPER(after(s(z0), x1)) → c20(AFTER(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 158.85/58.18
PROPER(after(after(z0, z1), x1)) → c20(AFTER(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1)) 158.85/58.18
PROPER(after(x0, 0)) → c20(AFTER(proper(x0), ok(0)), PROPER(x0)) 158.85/58.18
PROPER(after(0, x1)) → c20(AFTER(ok(0), proper(x1)), PROPER(x1)) 158.85/58.18
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0))) 158.85/58.18
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.18
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0))) 158.85/58.18
TOP(mark(after(z0, z1))) → c22(TOP(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.18
TOP(mark(0)) → c22(TOP(ok(0)))
S tuples:

ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2)) 158.85/58.18
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0)) 158.85/58.18
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1)) 158.85/58.18
FROM(mark(z0)) → c8(FROM(z0)) 158.85/58.18
FROM(ok(z0)) → c9(FROM(z0)) 158.85/58.18
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.85/58.18
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.85/58.18
S(mark(z0)) → c12(S(z0)) 158.85/58.18
S(ok(z0)) → c13(S(z0)) 158.85/58.18
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.85/58.18
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.85/58.18
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1)) 158.85/58.18
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0)) 158.85/58.18
ACTIVE(from(z0)) → c(S(z0)) 158.85/58.18
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.85/58.18
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.18
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.18
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 158.85/58.18
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.18
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 158.85/58.18
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.18
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.85/58.18
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 158.85/58.18
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0))) 158.85/58.18
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.18
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 158.85/58.18
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 158.85/58.18
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 158.85/58.18
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1))) 158.85/58.18
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1))) 158.85/58.18
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.85/58.18
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.18
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.19
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 158.85/58.19
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.19
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 158.85/58.19
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.19
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.85/58.19
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 158.85/58.19
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.19
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 158.85/58.19
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.19
PROPER(from(0)) → c17(FROM(ok(0))) 158.85/58.19
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 158.85/58.19
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 158.85/58.19
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 158.85/58.19
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1))) 158.85/58.19
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 158.85/58.19
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 158.85/58.19
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 158.85/58.19
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1)) 158.85/58.19
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0)) 158.85/58.19
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1)) 158.85/58.19
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0))) 158.85/58.19
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.19
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0))) 158.85/58.19
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.19
PROPER(s(0)) → c19(S(ok(0))) 158.85/58.19
PROPER(after(x0, from(z0))) → c20(AFTER(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 158.85/58.19
PROPER(after(x0, cons(z0, z1))) → c20(AFTER(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 158.85/58.19
PROPER(after(x0, s(z0))) → c20(AFTER(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 158.85/58.19
PROPER(after(x0, after(z0, z1))) → c20(AFTER(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1))) 158.85/58.19
PROPER(after(from(z0), x1)) → c20(AFTER(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 158.85/58.19
PROPER(after(cons(z0, z1), x1)) → c20(AFTER(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 158.85/58.19
PROPER(after(s(z0), x1)) → c20(AFTER(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 158.85/58.19
PROPER(after(after(z0, z1), x1)) → c20(AFTER(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1)) 158.85/58.19
PROPER(after(x0, 0)) → c20(AFTER(proper(x0), ok(0)), PROPER(x0)) 158.85/58.19
PROPER(after(0, x1)) → c20(AFTER(ok(0), proper(x1)), PROPER(x1)) 158.85/58.19
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0))) 158.85/58.19
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.19
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0))) 158.85/58.19
TOP(mark(after(z0, z1))) → c22(TOP(after(proper(z0), proper(z1))), PROPER(after(z0, z1)))
K tuples:

TOP(mark(0)) → c22(TOP(ok(0)))
Defined Rule Symbols:

active, from, cons, s, after, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, AFTER, 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

158.85/58.19
158.85/58.19

(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))) 158.85/58.19
TOP(ok(after(0, z0))) → c23(TOP(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.19
TOP(ok(after(s(z0), cons(z1, z2)))) → c23(TOP(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.19
TOP(ok(from(z0))) → c23(TOP(from(active(z0))), ACTIVE(from(z0))) 158.85/58.19
TOP(ok(cons(z0, z1))) → c23(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.19
TOP(ok(s(z0))) → c23(TOP(s(active(z0))), ACTIVE(s(z0))) 158.85/58.19
TOP(ok(after(z0, z1))) → c23(TOP(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.19
TOP(ok(after(z0, z1))) → c23(TOP(after(z0, active(z1))), ACTIVE(after(z0, z1)))
158.85/58.19
158.85/58.19

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 158.85/58.19
active(after(0, z0)) → mark(z0) 158.85/58.19
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2)) 158.85/58.19
active(from(z0)) → from(active(z0)) 158.85/58.19
active(cons(z0, z1)) → cons(active(z0), z1) 158.85/58.19
active(s(z0)) → s(active(z0)) 158.85/58.19
active(after(z0, z1)) → after(active(z0), z1) 158.85/58.19
active(after(z0, z1)) → after(z0, active(z1)) 158.85/58.19
from(mark(z0)) → mark(from(z0)) 158.85/58.19
from(ok(z0)) → ok(from(z0)) 158.85/58.19
cons(mark(z0), z1) → mark(cons(z0, z1)) 158.85/58.19
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 158.85/58.19
s(mark(z0)) → mark(s(z0)) 158.85/58.19
s(ok(z0)) → ok(s(z0)) 158.85/58.19
after(mark(z0), z1) → mark(after(z0, z1)) 158.85/58.19
after(z0, mark(z1)) → mark(after(z0, z1)) 158.85/58.19
after(ok(z0), ok(z1)) → ok(after(z0, z1)) 158.85/58.19
proper(from(z0)) → from(proper(z0)) 158.85/58.19
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 158.85/58.19
proper(s(z0)) → s(proper(z0)) 158.85/58.19
proper(after(z0, z1)) → after(proper(z0), proper(z1)) 158.85/58.19
proper(0) → ok(0) 158.85/58.19
top(mark(z0)) → top(proper(z0)) 158.85/58.19
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2)) 158.85/58.19
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0)) 158.85/58.19
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1)) 158.85/58.19
FROM(mark(z0)) → c8(FROM(z0)) 158.85/58.19
FROM(ok(z0)) → c9(FROM(z0)) 158.85/58.19
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.85/58.19
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.85/58.19
S(mark(z0)) → c12(S(z0)) 158.85/58.19
S(ok(z0)) → c13(S(z0)) 158.85/58.19
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.85/58.19
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.85/58.19
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1)) 158.85/58.19
ACTIVE(from(z0)) → c(S(z0)) 158.85/58.19
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.85/58.19
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.19
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.19
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 158.85/58.19
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.19
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 158.85/58.19
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.19
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.85/58.19
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 158.85/58.19
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0))) 158.85/58.19
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.19
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 158.85/58.19
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 158.85/58.19
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 158.85/58.19
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1))) 158.85/58.19
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1))) 158.85/58.19
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.85/58.19
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.19
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.19
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 158.85/58.19
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.19
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 158.85/58.19
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.19
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.85/58.19
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 158.85/58.19
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.19
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 158.85/58.19
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.19
PROPER(from(0)) → c17(FROM(ok(0))) 158.85/58.19
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 158.85/58.19
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 158.85/58.19
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 158.85/58.19
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1))) 158.85/58.19
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 158.85/58.19
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 158.85/58.19
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 158.85/58.19
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1)) 158.85/58.19
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0)) 158.85/58.19
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1)) 158.85/58.19
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0))) 158.85/58.19
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.19
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0))) 158.85/58.19
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.19
PROPER(s(0)) → c19(S(ok(0))) 158.85/58.19
PROPER(after(x0, from(z0))) → c20(AFTER(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 158.85/58.19
PROPER(after(x0, cons(z0, z1))) → c20(AFTER(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 158.85/58.19
PROPER(after(x0, s(z0))) → c20(AFTER(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 158.85/58.19
PROPER(after(x0, after(z0, z1))) → c20(AFTER(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1))) 158.85/58.19
PROPER(after(from(z0), x1)) → c20(AFTER(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 158.85/58.19
PROPER(after(cons(z0, z1), x1)) → c20(AFTER(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 158.85/58.19
PROPER(after(s(z0), x1)) → c20(AFTER(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 158.85/58.19
PROPER(after(after(z0, z1), x1)) → c20(AFTER(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1)) 158.85/58.19
PROPER(after(x0, 0)) → c20(AFTER(proper(x0), ok(0)), PROPER(x0)) 158.85/58.19
PROPER(after(0, x1)) → c20(AFTER(ok(0), proper(x1)), PROPER(x1)) 158.85/58.19
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0))) 158.85/58.19
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.19
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0))) 158.85/58.19
TOP(mark(after(z0, z1))) → c22(TOP(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.19
TOP(mark(0)) → c22(TOP(ok(0))) 158.85/58.19
TOP(ok(from(z0))) → c23(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.85/58.19
TOP(ok(after(0, z0))) → c23(TOP(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.19
TOP(ok(after(s(z0), cons(z1, z2)))) → c23(TOP(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.19
TOP(ok(from(z0))) → c23(TOP(from(active(z0))), ACTIVE(from(z0))) 158.85/58.19
TOP(ok(cons(z0, z1))) → c23(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.19
TOP(ok(s(z0))) → c23(TOP(s(active(z0))), ACTIVE(s(z0))) 158.85/58.19
TOP(ok(after(z0, z1))) → c23(TOP(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.19
TOP(ok(after(z0, z1))) → c23(TOP(after(z0, active(z1))), ACTIVE(after(z0, z1)))
S tuples:

ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2)) 158.85/58.19
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0)) 158.85/58.19
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1)) 158.85/58.19
FROM(mark(z0)) → c8(FROM(z0)) 158.85/58.19
FROM(ok(z0)) → c9(FROM(z0)) 158.85/58.19
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.85/58.19
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.85/58.19
S(mark(z0)) → c12(S(z0)) 158.85/58.19
S(ok(z0)) → c13(S(z0)) 158.85/58.19
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.85/58.19
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.85/58.19
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1)) 158.85/58.19
ACTIVE(from(z0)) → c(S(z0)) 158.85/58.19
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.85/58.19
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.19
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.19
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 158.85/58.19
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.19
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 158.85/58.19
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.19
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.85/58.19
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 158.85/58.19
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0))) 158.85/58.19
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.19
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 158.85/58.19
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 158.85/58.19
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 158.85/58.19
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1))) 158.85/58.19
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1))) 158.85/58.19
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.85/58.19
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.19
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.19
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 158.85/58.19
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.19
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 158.85/58.19
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.19
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.85/58.19
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 158.85/58.19
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.19
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 158.85/58.19
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.19
PROPER(from(0)) → c17(FROM(ok(0))) 158.85/58.19
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 158.85/58.19
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 158.85/58.19
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 158.85/58.19
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1))) 158.85/58.19
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 158.85/58.19
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 158.85/58.19
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 158.85/58.19
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1)) 158.85/58.19
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0)) 158.85/58.19
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1)) 158.85/58.19
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0))) 158.85/58.19
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.19
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0))) 158.85/58.19
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.19
PROPER(s(0)) → c19(S(ok(0))) 158.85/58.19
PROPER(after(x0, from(z0))) → c20(AFTER(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 158.85/58.19
PROPER(after(x0, cons(z0, z1))) → c20(AFTER(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 158.85/58.19
PROPER(after(x0, s(z0))) → c20(AFTER(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 158.85/58.19
PROPER(after(x0, after(z0, z1))) → c20(AFTER(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1))) 158.85/58.19
PROPER(after(from(z0), x1)) → c20(AFTER(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 158.85/58.19
PROPER(after(cons(z0, z1), x1)) → c20(AFTER(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 158.85/58.19
PROPER(after(s(z0), x1)) → c20(AFTER(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 158.85/58.19
PROPER(after(after(z0, z1), x1)) → c20(AFTER(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1)) 158.85/58.19
PROPER(after(x0, 0)) → c20(AFTER(proper(x0), ok(0)), PROPER(x0)) 158.85/58.19
PROPER(after(0, x1)) → c20(AFTER(ok(0), proper(x1)), PROPER(x1)) 158.85/58.19
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0))) 158.85/58.19
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.19
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0))) 158.85/58.19
TOP(mark(after(z0, z1))) → c22(TOP(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.19
TOP(ok(from(z0))) → c23(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.85/58.19
TOP(ok(after(0, z0))) → c23(TOP(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.19
TOP(ok(after(s(z0), cons(z1, z2)))) → c23(TOP(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.19
TOP(ok(from(z0))) → c23(TOP(from(active(z0))), ACTIVE(from(z0))) 158.85/58.19
TOP(ok(cons(z0, z1))) → c23(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.19
TOP(ok(s(z0))) → c23(TOP(s(active(z0))), ACTIVE(s(z0))) 158.85/58.19
TOP(ok(after(z0, z1))) → c23(TOP(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.19
TOP(ok(after(z0, z1))) → c23(TOP(after(z0, active(z1))), ACTIVE(after(z0, z1)))
K tuples:

TOP(mark(0)) → c22(TOP(ok(0)))
Defined Rule Symbols:

active, from, cons, s, after, proper, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, AFTER, 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

158.85/58.19
158.85/58.19

(35) CdtUnreachableProof (EQUIVALENT transformation)

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

ACTIVE(after(s(z0), cons(z1, z2))) → c2(AFTER(z0, z2)) 158.85/58.19
ACTIVE(after(z0, z1)) → c6(AFTER(active(z0), z1), ACTIVE(z0)) 158.85/58.19
ACTIVE(after(z0, z1)) → c7(AFTER(z0, active(z1)), ACTIVE(z1)) 158.85/58.19
ACTIVE(from(z0)) → c(S(z0)) 158.85/58.19
ACTIVE(from(from(z0))) → c3(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.85/58.19
ACTIVE(from(after(0, z0))) → c3(FROM(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.19
ACTIVE(from(after(s(z0), cons(z1, z2)))) → c3(FROM(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.19
ACTIVE(from(from(z0))) → c3(FROM(from(active(z0))), ACTIVE(from(z0))) 158.85/58.19
ACTIVE(from(cons(z0, z1))) → c3(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.19
ACTIVE(from(s(z0))) → c3(FROM(s(active(z0))), ACTIVE(s(z0))) 158.85/58.19
ACTIVE(from(after(z0, z1))) → c3(FROM(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.19
ACTIVE(from(after(z0, z1))) → c3(FROM(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.85/58.19
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 158.85/58.19
ACTIVE(cons(after(0, z0), x1)) → c4(CONS(mark(z0), x1), ACTIVE(after(0, z0))) 158.85/58.19
ACTIVE(cons(after(s(z0), cons(z1, z2)), x1)) → c4(CONS(mark(after(z0, z2)), x1), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.19
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 158.85/58.19
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 158.85/58.19
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 158.85/58.19
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(active(z0), z1), x1), ACTIVE(after(z0, z1))) 158.85/58.19
ACTIVE(cons(after(z0, z1), x1)) → c4(CONS(after(z0, active(z1)), x1), ACTIVE(after(z0, z1))) 158.85/58.19
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.85/58.19
ACTIVE(s(after(0, z0))) → c5(S(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.19
ACTIVE(s(after(s(z0), cons(z1, z2)))) → c5(S(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.19
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 158.85/58.19
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.19
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 158.85/58.19
ACTIVE(s(after(z0, z1))) → c5(S(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.19
ACTIVE(s(after(z0, z1))) → c5(S(after(z0, active(z1))), ACTIVE(after(z0, z1))) 158.85/58.19
PROPER(from(from(z0))) → c17(FROM(from(proper(z0))), PROPER(from(z0))) 158.85/58.19
PROPER(from(cons(z0, z1))) → c17(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.19
PROPER(from(s(z0))) → c17(FROM(s(proper(z0))), PROPER(s(z0))) 158.85/58.19
PROPER(from(after(z0, z1))) → c17(FROM(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.19
PROPER(from(0)) → c17(FROM(ok(0))) 158.85/58.19
PROPER(cons(x0, from(z0))) → c18(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 158.85/58.19
PROPER(cons(x0, cons(z0, z1))) → c18(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 158.85/58.19
PROPER(cons(x0, s(z0))) → c18(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 158.85/58.19
PROPER(cons(x0, after(z0, z1))) → c18(CONS(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1))) 158.85/58.19
PROPER(cons(from(z0), x1)) → c18(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 158.85/58.19
PROPER(cons(cons(z0, z1), x1)) → c18(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 158.85/58.19
PROPER(cons(s(z0), x1)) → c18(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 158.85/58.19
PROPER(cons(after(z0, z1), x1)) → c18(CONS(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1)) 158.85/58.19
PROPER(cons(x0, 0)) → c18(CONS(proper(x0), ok(0)), PROPER(x0)) 158.85/58.19
PROPER(cons(0, x1)) → c18(CONS(ok(0), proper(x1)), PROPER(x1)) 158.85/58.19
PROPER(s(from(z0))) → c19(S(from(proper(z0))), PROPER(from(z0))) 158.85/58.19
PROPER(s(cons(z0, z1))) → c19(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.19
PROPER(s(s(z0))) → c19(S(s(proper(z0))), PROPER(s(z0))) 158.85/58.19
PROPER(s(after(z0, z1))) → c19(S(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.19
PROPER(s(0)) → c19(S(ok(0))) 158.85/58.19
PROPER(after(x0, from(z0))) → c20(AFTER(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 158.85/58.19
PROPER(after(x0, cons(z0, z1))) → c20(AFTER(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 158.85/58.19
PROPER(after(x0, s(z0))) → c20(AFTER(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 158.85/58.19
PROPER(after(x0, after(z0, z1))) → c20(AFTER(proper(x0), after(proper(z0), proper(z1))), PROPER(x0), PROPER(after(z0, z1))) 158.85/58.19
PROPER(after(from(z0), x1)) → c20(AFTER(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 158.85/58.19
PROPER(after(cons(z0, z1), x1)) → c20(AFTER(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 158.85/58.19
PROPER(after(s(z0), x1)) → c20(AFTER(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 158.85/58.19
PROPER(after(after(z0, z1), x1)) → c20(AFTER(after(proper(z0), proper(z1)), proper(x1)), PROPER(after(z0, z1)), PROPER(x1)) 158.85/58.19
PROPER(after(x0, 0)) → c20(AFTER(proper(x0), ok(0)), PROPER(x0)) 158.85/58.19
PROPER(after(0, x1)) → c20(AFTER(ok(0), proper(x1)), PROPER(x1)) 158.85/58.19
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0))) 158.85/58.19
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 158.85/58.19
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0))) 158.85/58.19
TOP(mark(after(z0, z1))) → c22(TOP(after(proper(z0), proper(z1))), PROPER(after(z0, z1))) 158.85/58.19
TOP(ok(from(z0))) → c23(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 158.85/58.19
TOP(ok(after(0, z0))) → c23(TOP(mark(z0)), ACTIVE(after(0, z0))) 158.85/58.19
TOP(ok(after(s(z0), cons(z1, z2)))) → c23(TOP(mark(after(z0, z2))), ACTIVE(after(s(z0), cons(z1, z2)))) 158.85/58.19
TOP(ok(from(z0))) → c23(TOP(from(active(z0))), ACTIVE(from(z0))) 158.85/58.19
TOP(ok(cons(z0, z1))) → c23(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 158.85/58.19
TOP(ok(s(z0))) → c23(TOP(s(active(z0))), ACTIVE(s(z0))) 158.85/58.19
TOP(ok(after(z0, z1))) → c23(TOP(after(active(z0), z1)), ACTIVE(after(z0, z1))) 158.85/58.19
TOP(ok(after(z0, z1))) → c23(TOP(after(z0, active(z1))), ACTIVE(after(z0, z1)))
158.85/58.19
158.85/58.19

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 158.85/58.19
active(after(0, z0)) → mark(z0) 158.85/58.19
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2)) 158.85/58.19
active(from(z0)) → from(active(z0)) 158.85/58.19
active(cons(z0, z1)) → cons(active(z0), z1) 158.85/58.19
active(s(z0)) → s(active(z0)) 158.85/58.19
active(after(z0, z1)) → after(active(z0), z1) 158.85/58.19
active(after(z0, z1)) → after(z0, active(z1)) 158.85/58.19
from(mark(z0)) → mark(from(z0)) 158.85/58.19
from(ok(z0)) → ok(from(z0)) 158.85/58.19
cons(mark(z0), z1) → mark(cons(z0, z1)) 158.85/58.19
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 158.85/58.19
s(mark(z0)) → mark(s(z0)) 158.85/58.19
s(ok(z0)) → ok(s(z0)) 158.85/58.19
after(mark(z0), z1) → mark(after(z0, z1)) 158.85/58.19
after(z0, mark(z1)) → mark(after(z0, z1)) 158.85/58.19
after(ok(z0), ok(z1)) → ok(after(z0, z1)) 158.85/58.19
proper(from(z0)) → from(proper(z0)) 158.85/58.19
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 158.85/58.19
proper(s(z0)) → s(proper(z0)) 158.85/58.19
proper(after(z0, z1)) → after(proper(z0), proper(z1)) 158.85/58.19
proper(0) → ok(0) 158.85/58.19
top(mark(z0)) → top(proper(z0)) 158.85/58.19
top(ok(z0)) → top(active(z0))
Tuples:

FROM(mark(z0)) → c8(FROM(z0)) 158.85/58.19
FROM(ok(z0)) → c9(FROM(z0)) 158.85/58.19
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.85/58.19
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.85/58.19
S(mark(z0)) → c12(S(z0)) 158.85/58.19
S(ok(z0)) → c13(S(z0)) 158.85/58.19
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.85/58.19
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.85/58.19
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1)) 158.85/58.19
TOP(mark(0)) → c22(TOP(ok(0)))
S tuples:

FROM(mark(z0)) → c8(FROM(z0)) 158.85/58.19
FROM(ok(z0)) → c9(FROM(z0)) 158.85/58.19
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.85/58.19
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.85/58.19
S(mark(z0)) → c12(S(z0)) 158.85/58.19
S(ok(z0)) → c13(S(z0)) 158.85/58.19
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.85/58.19
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.85/58.19
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
K tuples:

TOP(mark(0)) → c22(TOP(ok(0)))
Defined Rule Symbols:

active, from, cons, s, after, proper, top

Defined Pair Symbols:

FROM, CONS, S, AFTER, TOP

Compound Symbols:

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

158.85/58.19
158.85/58.19

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

Removed 1 trailing tuple parts
158.85/58.19
158.85/58.19

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 158.85/58.19
active(after(0, z0)) → mark(z0) 158.85/58.19
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2)) 158.85/58.19
active(from(z0)) → from(active(z0)) 158.85/58.19
active(cons(z0, z1)) → cons(active(z0), z1) 158.85/58.19
active(s(z0)) → s(active(z0)) 158.85/58.19
active(after(z0, z1)) → after(active(z0), z1) 158.85/58.19
active(after(z0, z1)) → after(z0, active(z1)) 158.85/58.19
from(mark(z0)) → mark(from(z0)) 158.85/58.19
from(ok(z0)) → ok(from(z0)) 158.85/58.19
cons(mark(z0), z1) → mark(cons(z0, z1)) 158.85/58.19
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 158.85/58.19
s(mark(z0)) → mark(s(z0)) 158.85/58.19
s(ok(z0)) → ok(s(z0)) 158.85/58.19
after(mark(z0), z1) → mark(after(z0, z1)) 158.85/58.19
after(z0, mark(z1)) → mark(after(z0, z1)) 158.85/58.19
after(ok(z0), ok(z1)) → ok(after(z0, z1)) 158.85/58.19
proper(from(z0)) → from(proper(z0)) 158.85/58.19
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 158.85/58.19
proper(s(z0)) → s(proper(z0)) 158.85/58.19
proper(after(z0, z1)) → after(proper(z0), proper(z1)) 158.85/58.19
proper(0) → ok(0) 158.85/58.19
top(mark(z0)) → top(proper(z0)) 158.85/58.19
top(ok(z0)) → top(active(z0))
Tuples:

FROM(mark(z0)) → c8(FROM(z0)) 158.85/58.19
FROM(ok(z0)) → c9(FROM(z0)) 158.85/58.19
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.85/58.19
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.85/58.19
S(mark(z0)) → c12(S(z0)) 158.85/58.19
S(ok(z0)) → c13(S(z0)) 158.85/58.19
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.85/58.19
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.85/58.19
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1)) 158.85/58.19
TOP(mark(0)) → c22
S tuples:

FROM(mark(z0)) → c8(FROM(z0)) 158.85/58.19
FROM(ok(z0)) → c9(FROM(z0)) 158.85/58.19
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.85/58.20
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.85/58.20
S(mark(z0)) → c12(S(z0)) 158.85/58.20
S(ok(z0)) → c13(S(z0)) 158.85/58.20
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.85/58.20
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.85/58.20
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
K tuples:

TOP(mark(0)) → c22
Defined Rule Symbols:

active, from, cons, s, after, proper, top

Defined Pair Symbols:

FROM, CONS, S, AFTER, TOP

Compound Symbols:

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

158.85/58.20
158.85/58.20

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

Removed 1 trailing nodes:

TOP(mark(0)) → c22
158.85/58.20
158.85/58.20

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 158.85/58.20
active(after(0, z0)) → mark(z0) 158.85/58.20
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2)) 158.85/58.20
active(from(z0)) → from(active(z0)) 158.85/58.20
active(cons(z0, z1)) → cons(active(z0), z1) 158.85/58.20
active(s(z0)) → s(active(z0)) 158.85/58.20
active(after(z0, z1)) → after(active(z0), z1) 158.85/58.20
active(after(z0, z1)) → after(z0, active(z1)) 158.85/58.20
from(mark(z0)) → mark(from(z0)) 158.85/58.20
from(ok(z0)) → ok(from(z0)) 158.85/58.20
cons(mark(z0), z1) → mark(cons(z0, z1)) 158.85/58.20
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 158.85/58.20
s(mark(z0)) → mark(s(z0)) 158.85/58.20
s(ok(z0)) → ok(s(z0)) 158.85/58.20
after(mark(z0), z1) → mark(after(z0, z1)) 158.85/58.20
after(z0, mark(z1)) → mark(after(z0, z1)) 158.85/58.20
after(ok(z0), ok(z1)) → ok(after(z0, z1)) 158.85/58.20
proper(from(z0)) → from(proper(z0)) 158.85/58.20
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 158.85/58.20
proper(s(z0)) → s(proper(z0)) 158.85/58.20
proper(after(z0, z1)) → after(proper(z0), proper(z1)) 158.85/58.20
proper(0) → ok(0) 158.85/58.20
top(mark(z0)) → top(proper(z0)) 158.85/58.20
top(ok(z0)) → top(active(z0))
Tuples:

FROM(mark(z0)) → c8(FROM(z0)) 158.85/58.20
FROM(ok(z0)) → c9(FROM(z0)) 158.85/58.20
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.85/58.20
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.85/58.20
S(mark(z0)) → c12(S(z0)) 158.85/58.20
S(ok(z0)) → c13(S(z0)) 158.85/58.20
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.85/58.20
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.85/58.20
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
S tuples:

FROM(mark(z0)) → c8(FROM(z0)) 158.85/58.20
FROM(ok(z0)) → c9(FROM(z0)) 158.85/58.20
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.85/58.20
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.85/58.20
S(mark(z0)) → c12(S(z0)) 158.85/58.20
S(ok(z0)) → c13(S(z0)) 158.85/58.20
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.85/58.20
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.85/58.20
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
K tuples:none
Defined Rule Symbols:

active, from, cons, s, after, proper, top

Defined Pair Symbols:

FROM, CONS, S, AFTER

Compound Symbols:

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

158.85/58.20
158.85/58.20

(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)) 158.85/58.20
S(ok(z0)) → c13(S(z0)) 158.85/58.20
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.85/58.20
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.85/58.20
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

FROM(mark(z0)) → c8(FROM(z0)) 158.85/58.20
FROM(ok(z0)) → c9(FROM(z0)) 158.85/58.20
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.85/58.20
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.85/58.20
S(mark(z0)) → c12(S(z0)) 158.85/58.20
S(ok(z0)) → c13(S(z0)) 158.85/58.20
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.85/58.20
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.85/58.20
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 158.85/58.20

POL(AFTER(x1, x2)) = x1 + x2    158.85/58.20
POL(CONS(x1, x2)) = 0    158.85/58.20
POL(FROM(x1)) = 0    158.85/58.20
POL(S(x1)) = x1    158.85/58.20
POL(c10(x1)) = x1    158.85/58.20
POL(c11(x1)) = x1    158.85/58.20
POL(c12(x1)) = x1    158.85/58.20
POL(c13(x1)) = x1    158.85/58.20
POL(c14(x1)) = x1    158.85/58.20
POL(c15(x1)) = x1    158.85/58.20
POL(c16(x1)) = x1    158.85/58.20
POL(c8(x1)) = x1    158.85/58.20
POL(c9(x1)) = x1    158.85/58.20
POL(mark(x1)) = [1] + x1    158.85/58.20
POL(ok(x1)) = [1] + x1   
158.85/58.20
158.85/58.20

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 158.85/58.20
active(after(0, z0)) → mark(z0) 158.85/58.20
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2)) 158.85/58.20
active(from(z0)) → from(active(z0)) 158.85/58.20
active(cons(z0, z1)) → cons(active(z0), z1) 158.85/58.20
active(s(z0)) → s(active(z0)) 158.85/58.20
active(after(z0, z1)) → after(active(z0), z1) 158.85/58.20
active(after(z0, z1)) → after(z0, active(z1)) 158.85/58.20
from(mark(z0)) → mark(from(z0)) 158.85/58.20
from(ok(z0)) → ok(from(z0)) 158.85/58.20
cons(mark(z0), z1) → mark(cons(z0, z1)) 158.85/58.20
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 158.85/58.20
s(mark(z0)) → mark(s(z0)) 158.85/58.20
s(ok(z0)) → ok(s(z0)) 158.85/58.20
after(mark(z0), z1) → mark(after(z0, z1)) 158.85/58.20
after(z0, mark(z1)) → mark(after(z0, z1)) 158.85/58.20
after(ok(z0), ok(z1)) → ok(after(z0, z1)) 158.85/58.20
proper(from(z0)) → from(proper(z0)) 158.85/58.20
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 158.85/58.20
proper(s(z0)) → s(proper(z0)) 158.85/58.20
proper(after(z0, z1)) → after(proper(z0), proper(z1)) 158.85/58.20
proper(0) → ok(0) 158.85/58.20
top(mark(z0)) → top(proper(z0)) 158.85/58.20
top(ok(z0)) → top(active(z0))
Tuples:

FROM(mark(z0)) → c8(FROM(z0)) 158.85/58.20
FROM(ok(z0)) → c9(FROM(z0)) 158.85/58.20
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.85/58.20
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.85/58.20
S(mark(z0)) → c12(S(z0)) 158.85/58.20
S(ok(z0)) → c13(S(z0)) 158.85/58.20
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.85/58.20
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.85/58.20
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
S tuples:

FROM(mark(z0)) → c8(FROM(z0)) 158.85/58.20
FROM(ok(z0)) → c9(FROM(z0)) 158.85/58.20
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.85/58.20
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
K tuples:

S(mark(z0)) → c12(S(z0)) 158.85/58.20
S(ok(z0)) → c13(S(z0)) 158.85/58.20
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.85/58.20
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.85/58.20
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
Defined Rule Symbols:

active, from, cons, s, after, proper, top

Defined Pair Symbols:

FROM, CONS, S, AFTER

Compound Symbols:

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

158.85/58.20
158.85/58.20

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

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

FROM(mark(z0)) → c8(FROM(z0)) 158.85/58.20
CONS(mark(z0), z1) → c10(CONS(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

FROM(mark(z0)) → c8(FROM(z0)) 158.85/58.20
FROM(ok(z0)) → c9(FROM(z0)) 158.85/58.20
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.85/58.20
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.85/58.20
S(mark(z0)) → c12(S(z0)) 158.85/58.20
S(ok(z0)) → c13(S(z0)) 158.85/58.20
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.85/58.20
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.85/58.20
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 158.85/58.20

POL(AFTER(x1, x2)) = x2 + x22 + x1·x2 + x12 + x13 + x12·x2 + x1·x22 + x23    158.85/58.20
POL(CONS(x1, x2)) = x22 + x1·x2 + x12 + x13 + x12·x2 + x1·x22 + x23    158.85/58.20
POL(FROM(x1)) = x1 + x12 + x13    158.85/58.20
POL(S(x1)) = x1 + x12 + x13    158.85/58.20
POL(c10(x1)) = x1    158.85/58.20
POL(c11(x1)) = x1    158.85/58.20
POL(c12(x1)) = x1    158.85/58.20
POL(c13(x1)) = x1    158.85/58.20
POL(c14(x1)) = x1    158.85/58.20
POL(c15(x1)) = x1    158.85/58.20
POL(c16(x1)) = x1    158.85/58.20
POL(c8(x1)) = x1    158.85/58.20
POL(c9(x1)) = x1    158.85/58.20
POL(mark(x1)) = [1] + x1    158.85/58.20
POL(ok(x1)) = x1   
158.85/58.20
158.85/58.20

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 158.85/58.20
active(after(0, z0)) → mark(z0) 158.85/58.20
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2)) 158.85/58.20
active(from(z0)) → from(active(z0)) 158.85/58.20
active(cons(z0, z1)) → cons(active(z0), z1) 158.85/58.20
active(s(z0)) → s(active(z0)) 158.85/58.20
active(after(z0, z1)) → after(active(z0), z1) 158.85/58.20
active(after(z0, z1)) → after(z0, active(z1)) 158.85/58.20
from(mark(z0)) → mark(from(z0)) 158.85/58.20
from(ok(z0)) → ok(from(z0)) 158.85/58.20
cons(mark(z0), z1) → mark(cons(z0, z1)) 158.85/58.20
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 158.85/58.20
s(mark(z0)) → mark(s(z0)) 158.85/58.20
s(ok(z0)) → ok(s(z0)) 158.85/58.20
after(mark(z0), z1) → mark(after(z0, z1)) 158.85/58.20
after(z0, mark(z1)) → mark(after(z0, z1)) 158.85/58.20
after(ok(z0), ok(z1)) → ok(after(z0, z1)) 158.85/58.20
proper(from(z0)) → from(proper(z0)) 158.85/58.20
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 158.85/58.20
proper(s(z0)) → s(proper(z0)) 158.85/58.20
proper(after(z0, z1)) → after(proper(z0), proper(z1)) 158.85/58.20
proper(0) → ok(0) 158.85/58.20
top(mark(z0)) → top(proper(z0)) 158.85/58.20
top(ok(z0)) → top(active(z0))
Tuples:

FROM(mark(z0)) → c8(FROM(z0)) 158.85/58.20
FROM(ok(z0)) → c9(FROM(z0)) 158.85/58.20
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.85/58.20
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.85/58.20
S(mark(z0)) → c12(S(z0)) 158.85/58.20
S(ok(z0)) → c13(S(z0)) 158.85/58.20
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.85/58.20
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.85/58.20
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
S tuples:

FROM(ok(z0)) → c9(FROM(z0)) 158.85/58.20
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
K tuples:

S(mark(z0)) → c12(S(z0)) 158.85/58.20
S(ok(z0)) → c13(S(z0)) 158.85/58.20
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.85/58.20
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.85/58.20
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1)) 158.85/58.20
FROM(mark(z0)) → c8(FROM(z0)) 158.85/58.20
CONS(mark(z0), z1) → c10(CONS(z0, z1))
Defined Rule Symbols:

active, from, cons, s, after, proper, top

Defined Pair Symbols:

FROM, CONS, S, AFTER

Compound Symbols:

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

158.85/58.20
158.85/58.20

(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(ok(z0), ok(z1)) → c11(CONS(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

FROM(mark(z0)) → c8(FROM(z0)) 158.85/58.20
FROM(ok(z0)) → c9(FROM(z0)) 158.85/58.20
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.85/58.20
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.85/58.20
S(mark(z0)) → c12(S(z0)) 158.85/58.20
S(ok(z0)) → c13(S(z0)) 158.85/58.20
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.85/58.20
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.85/58.20
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 158.85/58.20

POL(AFTER(x1, x2)) = [5]x1 + [3]x2    158.85/58.20
POL(CONS(x1, x2)) = x1    158.85/58.20
POL(FROM(x1)) = 0    158.85/58.20
POL(S(x1)) = [3]x1    158.85/58.20
POL(c10(x1)) = x1    158.85/58.20
POL(c11(x1)) = x1    158.85/58.20
POL(c12(x1)) = x1    158.85/58.20
POL(c13(x1)) = x1    158.85/58.20
POL(c14(x1)) = x1    158.85/58.20
POL(c15(x1)) = x1    158.85/58.20
POL(c16(x1)) = x1    158.85/58.20
POL(c8(x1)) = x1    158.85/58.20
POL(c9(x1)) = x1    158.85/58.20
POL(mark(x1)) = x1    158.85/58.20
POL(ok(x1)) = [1] + x1   
158.85/58.20
158.85/58.20

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 158.85/58.20
active(after(0, z0)) → mark(z0) 158.85/58.20
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2)) 158.85/58.20
active(from(z0)) → from(active(z0)) 158.85/58.20
active(cons(z0, z1)) → cons(active(z0), z1) 158.85/58.20
active(s(z0)) → s(active(z0)) 158.85/58.20
active(after(z0, z1)) → after(active(z0), z1) 158.85/58.20
active(after(z0, z1)) → after(z0, active(z1)) 158.85/58.20
from(mark(z0)) → mark(from(z0)) 158.85/58.20
from(ok(z0)) → ok(from(z0)) 158.85/58.20
cons(mark(z0), z1) → mark(cons(z0, z1)) 158.85/58.20
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 158.85/58.20
s(mark(z0)) → mark(s(z0)) 158.85/58.20
s(ok(z0)) → ok(s(z0)) 158.85/58.20
after(mark(z0), z1) → mark(after(z0, z1)) 158.85/58.20
after(z0, mark(z1)) → mark(after(z0, z1)) 158.85/58.20
after(ok(z0), ok(z1)) → ok(after(z0, z1)) 158.85/58.20
proper(from(z0)) → from(proper(z0)) 158.85/58.20
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 158.85/58.20
proper(s(z0)) → s(proper(z0)) 158.85/58.20
proper(after(z0, z1)) → after(proper(z0), proper(z1)) 158.85/58.20
proper(0) → ok(0) 158.85/58.20
top(mark(z0)) → top(proper(z0)) 158.85/58.20
top(ok(z0)) → top(active(z0))
Tuples:

FROM(mark(z0)) → c8(FROM(z0)) 158.85/58.20
FROM(ok(z0)) → c9(FROM(z0)) 158.85/58.20
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.85/58.20
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.85/58.20
S(mark(z0)) → c12(S(z0)) 158.85/58.20
S(ok(z0)) → c13(S(z0)) 158.85/58.20
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.85/58.20
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.85/58.20
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
S tuples:

FROM(ok(z0)) → c9(FROM(z0))
K tuples:

S(mark(z0)) → c12(S(z0)) 158.85/58.20
S(ok(z0)) → c13(S(z0)) 158.85/58.20
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.85/58.20
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.85/58.20
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1)) 158.85/58.20
FROM(mark(z0)) → c8(FROM(z0)) 158.85/58.20
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.85/58.20
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1))
Defined Rule Symbols:

active, from, cons, s, after, proper, top

Defined Pair Symbols:

FROM, CONS, S, AFTER

Compound Symbols:

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

158.85/58.20
158.85/58.20

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

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

FROM(ok(z0)) → c9(FROM(z0))
We considered the (Usable) Rules:none
And the Tuples:

FROM(mark(z0)) → c8(FROM(z0)) 158.85/58.20
FROM(ok(z0)) → c9(FROM(z0)) 158.85/58.20
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.85/58.20
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.85/58.20
S(mark(z0)) → c12(S(z0)) 158.85/58.20
S(ok(z0)) → c13(S(z0)) 158.85/58.20
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.85/58.20
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.85/58.20
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 158.85/58.20

POL(AFTER(x1, x2)) = [3]x1 + [3]x2    158.85/58.20
POL(CONS(x1, x2)) = [3]x1 + [5]x2    158.85/58.20
POL(FROM(x1)) = [3]x1    158.85/58.20
POL(S(x1)) = [5]x1    158.85/58.20
POL(c10(x1)) = x1    158.85/58.20
POL(c11(x1)) = x1    158.85/58.20
POL(c12(x1)) = x1    158.85/58.20
POL(c13(x1)) = x1    158.85/58.20
POL(c14(x1)) = x1    158.85/58.20
POL(c15(x1)) = x1    158.85/58.20
POL(c16(x1)) = x1    158.85/58.20
POL(c8(x1)) = x1    158.85/58.20
POL(c9(x1)) = x1    158.85/58.20
POL(mark(x1)) = x1    158.85/58.20
POL(ok(x1)) = [1] + x1   
158.85/58.20
158.85/58.20

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0)))) 158.85/58.20
active(after(0, z0)) → mark(z0) 158.85/58.20
active(after(s(z0), cons(z1, z2))) → mark(after(z0, z2)) 158.85/58.20
active(from(z0)) → from(active(z0)) 158.85/58.20
active(cons(z0, z1)) → cons(active(z0), z1) 158.85/58.20
active(s(z0)) → s(active(z0)) 158.85/58.20
active(after(z0, z1)) → after(active(z0), z1) 158.85/58.20
active(after(z0, z1)) → after(z0, active(z1)) 158.85/58.20
from(mark(z0)) → mark(from(z0)) 158.85/58.20
from(ok(z0)) → ok(from(z0)) 158.85/58.20
cons(mark(z0), z1) → mark(cons(z0, z1)) 158.85/58.20
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 158.85/58.20
s(mark(z0)) → mark(s(z0)) 158.85/58.20
s(ok(z0)) → ok(s(z0)) 158.85/58.20
after(mark(z0), z1) → mark(after(z0, z1)) 158.85/58.20
after(z0, mark(z1)) → mark(after(z0, z1)) 158.85/58.20
after(ok(z0), ok(z1)) → ok(after(z0, z1)) 158.85/58.20
proper(from(z0)) → from(proper(z0)) 158.85/58.20
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 158.85/58.20
proper(s(z0)) → s(proper(z0)) 158.85/58.20
proper(after(z0, z1)) → after(proper(z0), proper(z1)) 158.85/58.20
proper(0) → ok(0) 158.85/58.20
top(mark(z0)) → top(proper(z0)) 158.85/58.20
top(ok(z0)) → top(active(z0))
Tuples:

FROM(mark(z0)) → c8(FROM(z0)) 158.85/58.20
FROM(ok(z0)) → c9(FROM(z0)) 158.85/58.20
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.85/58.20
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.85/58.20
S(mark(z0)) → c12(S(z0)) 158.85/58.20
S(ok(z0)) → c13(S(z0)) 158.85/58.20
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.85/58.20
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.85/58.20
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1))
S tuples:none
K tuples:

S(mark(z0)) → c12(S(z0)) 158.85/58.20
S(ok(z0)) → c13(S(z0)) 158.85/58.20
AFTER(mark(z0), z1) → c14(AFTER(z0, z1)) 158.85/58.20
AFTER(z0, mark(z1)) → c15(AFTER(z0, z1)) 158.85/58.20
AFTER(ok(z0), ok(z1)) → c16(AFTER(z0, z1)) 158.85/58.20
FROM(mark(z0)) → c8(FROM(z0)) 158.85/58.20
CONS(mark(z0), z1) → c10(CONS(z0, z1)) 158.85/58.20
CONS(ok(z0), ok(z1)) → c11(CONS(z0, z1)) 158.85/58.20
FROM(ok(z0)) → c9(FROM(z0))
Defined Rule Symbols:

active, from, cons, s, after, proper, top

Defined Pair Symbols:

FROM, CONS, S, AFTER

Compound Symbols:

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

158.85/58.20
158.85/58.20

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

The set S is empty
158.85/58.20
158.85/58.20

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

158.85/58.20
158.85/58.20
159.23/58.29 EOF