YES(O(1), O(n^3)) 157.12/56.92 YES(O(1), O(n^3)) 157.12/56.97 157.12/56.97 157.12/56.97 157.12/56.97 157.12/56.97 157.12/56.97 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 157.12/56.97 157.12/56.97 157.12/56.97
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157.12/56.97
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^3)).

0 CpxTRS
157.12/56.97 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
157.12/56.97 
↳2 CdtProblem
157.12/56.97 
↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
157.12/56.97 
↳4 CdtProblem
157.12/56.97 
↳5 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
157.12/56.97 
↳6 CdtProblem
157.12/56.97 
↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
157.12/56.97 
↳8 CdtProblem
157.12/56.97 
↳9 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
157.12/56.97 
↳10 CdtProblem
157.12/56.97 
↳11 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
157.12/56.97 
↳12 CdtProblem
157.12/56.97 
↳13 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
157.12/56.97 
↳14 CdtProblem
157.12/56.97 
↳15 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
157.12/56.97 
↳16 CdtProblem
157.12/56.97 
↳17 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
157.12/56.97 
↳18 CdtProblem
157.12/56.97 
↳19 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
157.12/56.97 
↳20 CdtProblem
157.12/56.97 
↳21 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
157.12/56.97 
↳22 CdtProblem
157.12/56.97 
↳23 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
157.12/56.97 
↳24 CdtProblem
157.12/56.97 
↳25 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
157.12/56.97 
↳26 CdtProblem
157.12/56.97 
↳27 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
157.12/56.97 
↳28 CdtProblem
157.12/56.97 
↳29 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
157.12/56.97 
↳30 CdtProblem
157.12/56.97 
↳31 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
157.12/56.97 
↳32 CdtProblem
157.12/56.97 
↳33 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
157.12/56.97 
↳34 CdtProblem
157.12/56.97 
↳35 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
157.12/56.97 
↳36 CdtProblem
157.12/56.97 
↳37 CdtUnreachableProof (⇔)
157.12/56.97 
↳38 CdtProblem
157.12/56.97 
↳39 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
157.12/56.97 
↳40 CdtProblem
157.12/56.97 
↳41 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
157.12/56.97 
↳42 CdtProblem
157.12/56.97 
↳43 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
157.12/56.97 
↳44 CdtProblem
157.12/56.97 
↳45 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
157.12/56.97 
↳46 CdtProblem
157.12/56.97 
↳47 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^3))))
157.12/56.97 
↳48 CdtProblem
157.12/56.97 
↳49 SIsEmptyProof (BOTH BOUNDS(ID, ID))
157.12/56.97 
↳50 BOUNDS(O(1), O(1))
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157.12/56.97

(0) Obligation:

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

active(first(0, X)) → mark(nil) 157.12/56.97
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z))) 157.12/56.97
active(from(X)) → mark(cons(X, from(s(X)))) 157.12/56.97
active(first(X1, X2)) → first(active(X1), X2) 157.12/56.97
active(first(X1, X2)) → first(X1, active(X2)) 157.12/56.97
active(s(X)) → s(active(X)) 157.12/56.97
active(cons(X1, X2)) → cons(active(X1), X2) 157.12/56.97
active(from(X)) → from(active(X)) 157.12/56.97
first(mark(X1), X2) → mark(first(X1, X2)) 157.12/56.97
first(X1, mark(X2)) → mark(first(X1, X2)) 157.12/56.97
s(mark(X)) → mark(s(X)) 157.12/56.97
cons(mark(X1), X2) → mark(cons(X1, X2)) 157.12/56.97
from(mark(X)) → mark(from(X)) 157.12/56.97
proper(first(X1, X2)) → first(proper(X1), proper(X2)) 157.12/56.97
proper(0) → ok(0) 157.12/56.97
proper(nil) → ok(nil) 157.12/56.97
proper(s(X)) → s(proper(X)) 157.12/56.97
proper(cons(X1, X2)) → cons(proper(X1), proper(X2)) 157.12/56.97
proper(from(X)) → from(proper(X)) 157.12/56.97
first(ok(X1), ok(X2)) → ok(first(X1, X2)) 157.12/56.97
s(ok(X)) → ok(s(X)) 157.12/56.97
cons(ok(X1), ok(X2)) → ok(cons(X1, X2)) 157.12/56.97
from(ok(X)) → ok(from(X)) 157.12/56.97
top(mark(X)) → top(proper(X)) 157.12/56.97
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST
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157.12/56.97

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

Converted CpxTRS to CDT
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157.12/56.97

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil) 157.12/56.97
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 157.12/56.97
active(from(z0)) → mark(cons(z0, from(s(z0)))) 157.12/56.97
active(first(z0, z1)) → first(active(z0), z1) 157.12/56.97
active(first(z0, z1)) → first(z0, active(z1)) 157.12/56.97
active(s(z0)) → s(active(z0)) 157.12/56.97
active(cons(z0, z1)) → cons(active(z0), z1) 157.12/56.97
active(from(z0)) → from(active(z0)) 157.12/56.97
first(mark(z0), z1) → mark(first(z0, z1)) 157.12/56.97
first(z0, mark(z1)) → mark(first(z0, z1)) 157.12/56.97
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 157.12/56.97
s(mark(z0)) → mark(s(z0)) 157.12/56.97
s(ok(z0)) → ok(s(z0)) 157.12/56.97
cons(mark(z0), z1) → mark(cons(z0, z1)) 157.12/56.97
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 157.12/56.97
from(mark(z0)) → mark(from(z0)) 157.12/56.97
from(ok(z0)) → ok(from(z0)) 157.12/56.97
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 157.12/56.97
proper(0) → ok(0) 157.12/56.97
proper(nil) → ok(nil) 157.12/56.97
proper(s(z0)) → s(proper(z0)) 157.12/56.97
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 157.12/56.97
proper(from(z0)) → from(proper(z0)) 157.12/56.97
top(mark(z0)) → top(proper(z0)) 157.12/56.97
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)), FIRST(z0, z2)) 157.12/56.97
ACTIVE(from(z0)) → c2(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0)) 157.12/56.97
ACTIVE(first(z0, z1)) → c3(FIRST(active(z0), z1), ACTIVE(z0)) 157.12/56.97
ACTIVE(first(z0, z1)) → c4(FIRST(z0, active(z1)), ACTIVE(z1)) 157.12/56.97
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0)) 157.12/56.97
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0)) 157.12/56.97
ACTIVE(from(z0)) → c7(FROM(active(z0)), ACTIVE(z0)) 157.12/56.97
FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.12/56.97
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.12/56.97
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.12/56.97
S(mark(z0)) → c11(S(z0)) 157.12/56.97
S(ok(z0)) → c12(S(z0)) 157.12/56.97
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.12/56.97
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.12/56.97
FROM(mark(z0)) → c15(FROM(z0)) 157.12/56.98
FROM(ok(z0)) → c16(FROM(z0)) 157.12/56.98
PROPER(first(z0, z1)) → c17(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 157.12/56.98
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 157.12/56.98
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 157.12/56.98
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 157.12/56.98
TOP(mark(z0)) → c23(TOP(proper(z0)), PROPER(z0)) 157.12/56.98
TOP(ok(z0)) → c24(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)), FIRST(z0, z2)) 157.12/56.98
ACTIVE(from(z0)) → c2(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0)) 157.12/56.98
ACTIVE(first(z0, z1)) → c3(FIRST(active(z0), z1), ACTIVE(z0)) 157.12/56.98
ACTIVE(first(z0, z1)) → c4(FIRST(z0, active(z1)), ACTIVE(z1)) 157.12/56.98
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0)) 157.12/56.98
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0)) 157.12/56.98
ACTIVE(from(z0)) → c7(FROM(active(z0)), ACTIVE(z0)) 157.12/56.98
FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.12/56.98
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.12/56.98
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.12/56.98
S(mark(z0)) → c11(S(z0)) 157.12/56.98
S(ok(z0)) → c12(S(z0)) 157.12/56.98
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.12/56.98
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.12/56.98
FROM(mark(z0)) → c15(FROM(z0)) 157.12/56.98
FROM(ok(z0)) → c16(FROM(z0)) 157.12/56.98
PROPER(first(z0, z1)) → c17(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 157.12/56.98
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 157.12/56.98
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 157.12/56.98
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 157.12/56.98
TOP(mark(z0)) → c23(TOP(proper(z0)), PROPER(z0)) 157.12/56.98
TOP(ok(z0)) → c24(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:

active, first, s, cons, from, proper, top

Defined Pair Symbols:

ACTIVE, FIRST, S, CONS, FROM, PROPER, TOP

Compound Symbols:

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

157.12/56.98
157.12/56.98

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

Removed 2 trailing tuple parts
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157.12/56.98

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil) 157.12/56.98
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 157.12/56.98
active(from(z0)) → mark(cons(z0, from(s(z0)))) 157.12/56.98
active(first(z0, z1)) → first(active(z0), z1) 157.12/56.98
active(first(z0, z1)) → first(z0, active(z1)) 157.12/56.98
active(s(z0)) → s(active(z0)) 157.12/56.98
active(cons(z0, z1)) → cons(active(z0), z1) 157.12/56.98
active(from(z0)) → from(active(z0)) 157.12/56.98
first(mark(z0), z1) → mark(first(z0, z1)) 157.12/56.98
first(z0, mark(z1)) → mark(first(z0, z1)) 157.12/56.98
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 157.12/56.98
s(mark(z0)) → mark(s(z0)) 157.12/56.98
s(ok(z0)) → ok(s(z0)) 157.12/56.98
cons(mark(z0), z1) → mark(cons(z0, z1)) 157.12/56.98
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 157.12/56.98
from(mark(z0)) → mark(from(z0)) 157.12/56.98
from(ok(z0)) → ok(from(z0)) 157.12/56.98
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 157.12/56.98
proper(0) → ok(0) 157.12/56.98
proper(nil) → ok(nil) 157.12/56.98
proper(s(z0)) → s(proper(z0)) 157.12/56.98
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 157.12/56.98
proper(from(z0)) → from(proper(z0)) 157.12/56.98
top(mark(z0)) → top(proper(z0)) 157.12/56.98
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)), FIRST(z0, z2)) 157.12/56.98
ACTIVE(first(z0, z1)) → c3(FIRST(active(z0), z1), ACTIVE(z0)) 157.12/56.98
ACTIVE(first(z0, z1)) → c4(FIRST(z0, active(z1)), ACTIVE(z1)) 157.12/56.98
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0)) 157.12/56.98
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0)) 157.12/56.98
ACTIVE(from(z0)) → c7(FROM(active(z0)), ACTIVE(z0)) 157.12/56.98
FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.12/56.98
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.12/56.98
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.12/56.98
S(mark(z0)) → c11(S(z0)) 157.12/56.98
S(ok(z0)) → c12(S(z0)) 157.12/56.98
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.12/56.98
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.12/56.98
FROM(mark(z0)) → c15(FROM(z0)) 157.12/56.98
FROM(ok(z0)) → c16(FROM(z0)) 157.12/56.98
PROPER(first(z0, z1)) → c17(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 157.12/56.98
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 157.12/56.98
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 157.12/56.98
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 157.12/56.98
TOP(mark(z0)) → c23(TOP(proper(z0)), PROPER(z0)) 157.57/57.02
TOP(ok(z0)) → c24(TOP(active(z0)), ACTIVE(z0)) 157.57/57.02
ACTIVE(from(z0)) → c2(S(z0))
S tuples:

ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)), FIRST(z0, z2)) 157.57/57.02
ACTIVE(first(z0, z1)) → c3(FIRST(active(z0), z1), ACTIVE(z0)) 157.57/57.02
ACTIVE(first(z0, z1)) → c4(FIRST(z0, active(z1)), ACTIVE(z1)) 157.57/57.02
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0)) 157.57/57.02
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0)) 157.57/57.02
ACTIVE(from(z0)) → c7(FROM(active(z0)), ACTIVE(z0)) 157.57/57.02
FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.57/57.02
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.57/57.02
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.57/57.02
S(mark(z0)) → c11(S(z0)) 157.57/57.02
S(ok(z0)) → c12(S(z0)) 157.57/57.02
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.57/57.02
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.57/57.02
FROM(mark(z0)) → c15(FROM(z0)) 157.57/57.02
FROM(ok(z0)) → c16(FROM(z0)) 157.57/57.02
PROPER(first(z0, z1)) → c17(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 157.57/57.02
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 157.57/57.02
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 157.57/57.02
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 157.57/57.02
TOP(mark(z0)) → c23(TOP(proper(z0)), PROPER(z0)) 157.57/57.02
TOP(ok(z0)) → c24(TOP(active(z0)), ACTIVE(z0)) 157.57/57.02
ACTIVE(from(z0)) → c2(S(z0))
K tuples:none
Defined Rule Symbols:

active, first, s, cons, from, proper, top

Defined Pair Symbols:

ACTIVE, FIRST, S, CONS, FROM, PROPER, TOP

Compound Symbols:

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

157.57/57.02
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(5) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC
157.57/57.02
157.57/57.02

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil) 157.57/57.02
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 157.57/57.02
active(from(z0)) → mark(cons(z0, from(s(z0)))) 157.57/57.02
active(first(z0, z1)) → first(active(z0), z1) 157.57/57.02
active(first(z0, z1)) → first(z0, active(z1)) 157.57/57.02
active(s(z0)) → s(active(z0)) 157.57/57.02
active(cons(z0, z1)) → cons(active(z0), z1) 157.57/57.02
active(from(z0)) → from(active(z0)) 157.57/57.02
first(mark(z0), z1) → mark(first(z0, z1)) 157.57/57.02
first(z0, mark(z1)) → mark(first(z0, z1)) 157.57/57.02
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 157.57/57.02
s(mark(z0)) → mark(s(z0)) 157.57/57.02
s(ok(z0)) → ok(s(z0)) 157.57/57.02
cons(mark(z0), z1) → mark(cons(z0, z1)) 157.57/57.02
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 157.57/57.02
from(mark(z0)) → mark(from(z0)) 157.57/57.02
from(ok(z0)) → ok(from(z0)) 157.57/57.02
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 157.57/57.02
proper(0) → ok(0) 157.57/57.02
proper(nil) → ok(nil) 157.57/57.02
proper(s(z0)) → s(proper(z0)) 157.57/57.02
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 157.57/57.02
proper(from(z0)) → from(proper(z0)) 157.57/57.02
top(mark(z0)) → top(proper(z0)) 157.57/57.02
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(first(z0, z1)) → c3(FIRST(active(z0), z1), ACTIVE(z0)) 157.57/57.07
ACTIVE(first(z0, z1)) → c4(FIRST(z0, active(z1)), ACTIVE(z1)) 157.57/57.07
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0)) 157.57/57.07
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0)) 157.57/57.07
ACTIVE(from(z0)) → c7(FROM(active(z0)), ACTIVE(z0)) 157.57/57.07
FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.57/57.07
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.57/57.07
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.57/57.07
S(mark(z0)) → c11(S(z0)) 157.57/57.07
S(ok(z0)) → c12(S(z0)) 157.57/57.07
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.57/57.07
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.57/57.07
FROM(mark(z0)) → c15(FROM(z0)) 157.57/57.07
FROM(ok(z0)) → c16(FROM(z0)) 157.57/57.07
PROPER(first(z0, z1)) → c17(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 157.57/57.07
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 157.57/57.07
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 157.57/57.07
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 157.57/57.07
TOP(mark(z0)) → c23(TOP(proper(z0)), PROPER(z0)) 157.57/57.07
TOP(ok(z0)) → c24(TOP(active(z0)), ACTIVE(z0)) 157.57/57.07
ACTIVE(from(z0)) → c2(S(z0)) 157.57/57.07
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 157.57/57.07
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2))
S tuples:

ACTIVE(first(z0, z1)) → c3(FIRST(active(z0), z1), ACTIVE(z0)) 157.57/57.07
ACTIVE(first(z0, z1)) → c4(FIRST(z0, active(z1)), ACTIVE(z1)) 157.57/57.07
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0)) 157.57/57.07
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0)) 157.57/57.07
ACTIVE(from(z0)) → c7(FROM(active(z0)), ACTIVE(z0)) 157.57/57.07
FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.57/57.07
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.57/57.07
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.57/57.07
S(mark(z0)) → c11(S(z0)) 157.57/57.07
S(ok(z0)) → c12(S(z0)) 157.57/57.07
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.57/57.07
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.57/57.07
FROM(mark(z0)) → c15(FROM(z0)) 157.57/57.07
FROM(ok(z0)) → c16(FROM(z0)) 157.57/57.07
PROPER(first(z0, z1)) → c17(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 157.57/57.07
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 157.57/57.07
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 157.57/57.07
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 157.57/57.07
TOP(mark(z0)) → c23(TOP(proper(z0)), PROPER(z0)) 157.57/57.07
TOP(ok(z0)) → c24(TOP(active(z0)), ACTIVE(z0)) 157.57/57.07
ACTIVE(from(z0)) → c2(S(z0)) 157.57/57.07
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 157.57/57.07
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2))
K tuples:none
Defined Rule Symbols:

active, first, s, cons, from, proper, top

Defined Pair Symbols:

ACTIVE, FIRST, S, CONS, FROM, PROPER, TOP

Compound Symbols:

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

157.57/57.07
157.57/57.07

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

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

TOP(mark(z0)) → c23(TOP(proper(z0)), PROPER(z0))
We considered the (Usable) Rules:

first(z0, mark(z1)) → mark(first(z0, z1)) 157.57/57.07
first(mark(z0), z1) → mark(first(z0, z1)) 157.57/57.07
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 157.57/57.07
active(first(0, z0)) → mark(nil) 157.57/57.07
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 157.57/57.07
active(from(z0)) → mark(cons(z0, from(s(z0)))) 157.57/57.07
active(first(z0, z1)) → first(active(z0), z1) 157.57/57.07
active(first(z0, z1)) → first(z0, active(z1)) 157.57/57.07
active(s(z0)) → s(active(z0)) 157.57/57.07
active(cons(z0, z1)) → cons(active(z0), z1) 157.57/57.07
active(from(z0)) → from(active(z0)) 157.57/57.07
from(mark(z0)) → mark(from(z0)) 157.57/57.07
from(ok(z0)) → ok(from(z0)) 157.57/57.07
cons(mark(z0), z1) → mark(cons(z0, z1)) 157.57/57.07
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 157.57/57.07
s(mark(z0)) → mark(s(z0)) 157.57/57.07
s(ok(z0)) → ok(s(z0)) 157.57/57.07
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 157.57/57.07
proper(0) → ok(0) 157.57/57.07
proper(nil) → ok(nil) 157.57/57.07
proper(s(z0)) → s(proper(z0)) 157.57/57.07
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 157.57/57.07
proper(from(z0)) → from(proper(z0))
And the Tuples:

ACTIVE(first(z0, z1)) → c3(FIRST(active(z0), z1), ACTIVE(z0)) 157.57/57.07
ACTIVE(first(z0, z1)) → c4(FIRST(z0, active(z1)), ACTIVE(z1)) 157.57/57.07
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0)) 157.57/57.07
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0)) 157.57/57.07
ACTIVE(from(z0)) → c7(FROM(active(z0)), ACTIVE(z0)) 157.57/57.07
FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.57/57.07
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.57/57.07
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.57/57.07
S(mark(z0)) → c11(S(z0)) 157.57/57.07
S(ok(z0)) → c12(S(z0)) 157.57/57.07
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.57/57.07
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.57/57.07
FROM(mark(z0)) → c15(FROM(z0)) 157.57/57.07
FROM(ok(z0)) → c16(FROM(z0)) 157.57/57.07
PROPER(first(z0, z1)) → c17(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 157.57/57.07
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 157.57/57.07
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 157.57/57.07
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 157.57/57.07
TOP(mark(z0)) → c23(TOP(proper(z0)), PROPER(z0)) 157.57/57.07
TOP(ok(z0)) → c24(TOP(active(z0)), ACTIVE(z0)) 157.57/57.07
ACTIVE(from(z0)) → c2(S(z0)) 157.57/57.07
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 157.57/57.07
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2))
The order we found is given by the following interpretation:
Polynomial interpretation : 157.57/57.07

POL(0) = [2]    157.57/57.07
POL(ACTIVE(x1)) = 0    157.57/57.07
POL(CONS(x1, x2)) = 0    157.57/57.07
POL(FIRST(x1, x2)) = 0    157.57/57.07
POL(FROM(x1)) = 0    157.57/57.07
POL(PROPER(x1)) = 0    157.57/57.07
POL(S(x1)) = 0    157.57/57.07
POL(TOP(x1)) = [2]x1    157.57/57.07
POL(active(x1)) = x1    157.57/57.07
POL(c(x1)) = x1    157.57/57.07
POL(c10(x1)) = x1    157.57/57.07
POL(c11(x1)) = x1    157.57/57.07
POL(c12(x1)) = x1    157.57/57.07
POL(c13(x1)) = x1    157.57/57.07
POL(c14(x1)) = x1    157.57/57.07
POL(c15(x1)) = x1    157.57/57.07
POL(c16(x1)) = x1    157.57/57.07
POL(c17(x1, x2, x3)) = x1 + x2 + x3    157.57/57.07
POL(c2(x1)) = x1    157.57/57.07
POL(c20(x1, x2)) = x1 + x2    157.57/57.07
POL(c21(x1, x2, x3)) = x1 + x2 + x3    157.57/57.07
POL(c22(x1, x2)) = x1 + x2    157.57/57.07
POL(c23(x1, x2)) = x1 + x2    157.57/57.07
POL(c24(x1, x2)) = x1 + x2    157.57/57.07
POL(c3(x1, x2)) = x1 + x2    157.57/57.07
POL(c4(x1, x2)) = x1 + x2    157.57/57.07
POL(c5(x1, x2)) = x1 + x2    157.57/57.07
POL(c6(x1, x2)) = x1 + x2    157.57/57.07
POL(c7(x1, x2)) = x1 + x2    157.57/57.07
POL(c8(x1)) = x1    157.57/57.07
POL(c9(x1)) = x1    157.57/57.07
POL(cons(x1, x2)) = [1] + [5]x1    157.57/57.07
POL(first(x1, x2)) = [2]x1 + [5]x2    157.57/57.07
POL(from(x1)) = [4] + [5]x1    157.57/57.07
POL(mark(x1)) = [3] + x1    157.57/57.07
POL(nil) = [1]    157.57/57.07
POL(ok(x1)) = x1    157.57/57.07
POL(proper(x1)) = x1    157.57/57.07
POL(s(x1)) = [2]x1   
157.57/57.07
157.57/57.07

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil) 157.57/57.07
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 157.57/57.07
active(from(z0)) → mark(cons(z0, from(s(z0)))) 157.57/57.07
active(first(z0, z1)) → first(active(z0), z1) 157.57/57.07
active(first(z0, z1)) → first(z0, active(z1)) 157.57/57.07
active(s(z0)) → s(active(z0)) 157.57/57.07
active(cons(z0, z1)) → cons(active(z0), z1) 157.57/57.07
active(from(z0)) → from(active(z0)) 157.57/57.07
first(mark(z0), z1) → mark(first(z0, z1)) 157.57/57.07
first(z0, mark(z1)) → mark(first(z0, z1)) 157.57/57.07
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 157.57/57.07
s(mark(z0)) → mark(s(z0)) 157.57/57.07
s(ok(z0)) → ok(s(z0)) 157.57/57.07
cons(mark(z0), z1) → mark(cons(z0, z1)) 157.57/57.07
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 157.57/57.07
from(mark(z0)) → mark(from(z0)) 157.57/57.07
from(ok(z0)) → ok(from(z0)) 157.57/57.07
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 157.57/57.07
proper(0) → ok(0) 157.57/57.07
proper(nil) → ok(nil) 157.57/57.07
proper(s(z0)) → s(proper(z0)) 157.57/57.07
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 157.57/57.07
proper(from(z0)) → from(proper(z0)) 157.57/57.07
top(mark(z0)) → top(proper(z0)) 157.57/57.07
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(first(z0, z1)) → c3(FIRST(active(z0), z1), ACTIVE(z0)) 157.57/57.07
ACTIVE(first(z0, z1)) → c4(FIRST(z0, active(z1)), ACTIVE(z1)) 157.57/57.07
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0)) 157.57/57.07
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0)) 157.57/57.07
ACTIVE(from(z0)) → c7(FROM(active(z0)), ACTIVE(z0)) 157.57/57.07
FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.57/57.07
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.57/57.07
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.57/57.07
S(mark(z0)) → c11(S(z0)) 157.57/57.07
S(ok(z0)) → c12(S(z0)) 157.57/57.07
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.57/57.07
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.57/57.07
FROM(mark(z0)) → c15(FROM(z0)) 157.57/57.07
FROM(ok(z0)) → c16(FROM(z0)) 157.57/57.07
PROPER(first(z0, z1)) → c17(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 157.57/57.07
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 157.57/57.07
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 157.57/57.07
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 157.57/57.07
TOP(mark(z0)) → c23(TOP(proper(z0)), PROPER(z0)) 157.57/57.07
TOP(ok(z0)) → c24(TOP(active(z0)), ACTIVE(z0)) 157.57/57.07
ACTIVE(from(z0)) → c2(S(z0)) 157.57/57.07
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 157.57/57.07
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2))
S tuples:

ACTIVE(first(z0, z1)) → c3(FIRST(active(z0), z1), ACTIVE(z0)) 157.57/57.07
ACTIVE(first(z0, z1)) → c4(FIRST(z0, active(z1)), ACTIVE(z1)) 157.57/57.07
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0)) 157.57/57.07
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0)) 157.57/57.07
ACTIVE(from(z0)) → c7(FROM(active(z0)), ACTIVE(z0)) 157.57/57.07
FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.57/57.07
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.57/57.07
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.57/57.07
S(mark(z0)) → c11(S(z0)) 157.57/57.07
S(ok(z0)) → c12(S(z0)) 157.57/57.07
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.57/57.07
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.57/57.07
FROM(mark(z0)) → c15(FROM(z0)) 157.57/57.07
FROM(ok(z0)) → c16(FROM(z0)) 157.57/57.07
PROPER(first(z0, z1)) → c17(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 157.57/57.07
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 157.57/57.07
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 157.57/57.07
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 157.57/57.07
TOP(ok(z0)) → c24(TOP(active(z0)), ACTIVE(z0)) 157.57/57.07
ACTIVE(from(z0)) → c2(S(z0)) 157.57/57.07
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 157.57/57.07
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2))
K tuples:

TOP(mark(z0)) → c23(TOP(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, first, s, cons, from, proper, top

Defined Pair Symbols:

ACTIVE, FIRST, S, CONS, FROM, PROPER, TOP

Compound Symbols:

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

157.57/57.07
157.57/57.07

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

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

ACTIVE(s(first(0, z0))) → c5(S(mark(nil)), ACTIVE(first(0, z0))) 157.57/57.07
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c5(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.57/57.07
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.57/57.07
ACTIVE(s(first(z0, z1))) → c5(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.57/57.07
ACTIVE(s(first(z0, z1))) → c5(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.57/57.07
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 157.57/57.07
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.57/57.07
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
157.57/57.07
157.57/57.07

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil) 157.57/57.07
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 157.57/57.07
active(from(z0)) → mark(cons(z0, from(s(z0)))) 157.57/57.07
active(first(z0, z1)) → first(active(z0), z1) 157.57/57.07
active(first(z0, z1)) → first(z0, active(z1)) 157.57/57.07
active(s(z0)) → s(active(z0)) 157.57/57.07
active(cons(z0, z1)) → cons(active(z0), z1) 157.57/57.07
active(from(z0)) → from(active(z0)) 157.57/57.07
first(mark(z0), z1) → mark(first(z0, z1)) 157.57/57.07
first(z0, mark(z1)) → mark(first(z0, z1)) 157.57/57.07
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 157.57/57.07
s(mark(z0)) → mark(s(z0)) 157.57/57.07
s(ok(z0)) → ok(s(z0)) 157.57/57.07
cons(mark(z0), z1) → mark(cons(z0, z1)) 157.57/57.07
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 157.57/57.07
from(mark(z0)) → mark(from(z0)) 157.57/57.07
from(ok(z0)) → ok(from(z0)) 157.57/57.07
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 157.57/57.07
proper(0) → ok(0) 157.57/57.07
proper(nil) → ok(nil) 157.57/57.07
proper(s(z0)) → s(proper(z0)) 157.57/57.07
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 157.57/57.07
proper(from(z0)) → from(proper(z0)) 157.57/57.07
top(mark(z0)) → top(proper(z0)) 157.57/57.07
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(first(z0, z1)) → c3(FIRST(active(z0), z1), ACTIVE(z0)) 157.57/57.07
ACTIVE(first(z0, z1)) → c4(FIRST(z0, active(z1)), ACTIVE(z1)) 157.57/57.07
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0)) 157.57/57.07
ACTIVE(from(z0)) → c7(FROM(active(z0)), ACTIVE(z0)) 157.57/57.07
FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.57/57.07
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.57/57.07
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.57/57.07
S(mark(z0)) → c11(S(z0)) 157.57/57.07
S(ok(z0)) → c12(S(z0)) 157.57/57.07
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.57/57.07
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.57/57.07
FROM(mark(z0)) → c15(FROM(z0)) 157.57/57.07
FROM(ok(z0)) → c16(FROM(z0)) 157.57/57.07
PROPER(first(z0, z1)) → c17(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 157.57/57.07
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 157.57/57.09
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 157.57/57.09
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 157.57/57.09
TOP(mark(z0)) → c23(TOP(proper(z0)), PROPER(z0)) 157.57/57.09
TOP(ok(z0)) → c24(TOP(active(z0)), ACTIVE(z0)) 157.57/57.09
ACTIVE(from(z0)) → c2(S(z0)) 157.57/57.09
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 157.57/57.09
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 157.57/57.09
ACTIVE(s(first(0, z0))) → c5(S(mark(nil)), ACTIVE(first(0, z0))) 157.57/57.09
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c5(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.57/57.09
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.57/57.09
ACTIVE(s(first(z0, z1))) → c5(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.57/57.09
ACTIVE(s(first(z0, z1))) → c5(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.57/57.09
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 157.57/57.09
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.57/57.09
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
S tuples:

ACTIVE(first(z0, z1)) → c3(FIRST(active(z0), z1), ACTIVE(z0)) 157.57/57.09
ACTIVE(first(z0, z1)) → c4(FIRST(z0, active(z1)), ACTIVE(z1)) 157.57/57.09
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0)) 157.57/57.09
ACTIVE(from(z0)) → c7(FROM(active(z0)), ACTIVE(z0)) 157.57/57.09
FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.57/57.09
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.57/57.09
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.57/57.09
S(mark(z0)) → c11(S(z0)) 157.57/57.09
S(ok(z0)) → c12(S(z0)) 157.57/57.09
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.57/57.09
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.57/57.09
FROM(mark(z0)) → c15(FROM(z0)) 157.57/57.09
FROM(ok(z0)) → c16(FROM(z0)) 157.57/57.09
PROPER(first(z0, z1)) → c17(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 157.57/57.09
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 157.57/57.09
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 157.57/57.09
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 157.57/57.09
TOP(ok(z0)) → c24(TOP(active(z0)), ACTIVE(z0)) 157.57/57.09
ACTIVE(from(z0)) → c2(S(z0)) 157.57/57.09
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 157.57/57.09
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 157.57/57.09
ACTIVE(s(first(0, z0))) → c5(S(mark(nil)), ACTIVE(first(0, z0))) 157.57/57.09
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c5(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.57/57.09
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.57/57.09
ACTIVE(s(first(z0, z1))) → c5(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.57/57.09
ACTIVE(s(first(z0, z1))) → c5(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.57/57.09
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 157.57/57.09
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.57/57.09
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
K tuples:

TOP(mark(z0)) → c23(TOP(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, first, s, cons, from, proper, top

Defined Pair Symbols:

ACTIVE, FIRST, S, CONS, FROM, PROPER, TOP

Compound Symbols:

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

157.57/57.09
157.57/57.09

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

Use narrowing to replace ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0)) by

ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 157.57/57.09
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 157.57/57.09
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 157.57/57.09
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 157.57/57.09
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 157.57/57.09
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 157.57/57.09
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 157.57/57.09
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
157.57/57.09
157.57/57.09

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil) 157.57/57.09
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 157.57/57.09
active(from(z0)) → mark(cons(z0, from(s(z0)))) 157.57/57.09
active(first(z0, z1)) → first(active(z0), z1) 157.57/57.09
active(first(z0, z1)) → first(z0, active(z1)) 157.57/57.09
active(s(z0)) → s(active(z0)) 157.57/57.09
active(cons(z0, z1)) → cons(active(z0), z1) 157.57/57.09
active(from(z0)) → from(active(z0)) 157.57/57.09
first(mark(z0), z1) → mark(first(z0, z1)) 157.57/57.09
first(z0, mark(z1)) → mark(first(z0, z1)) 157.57/57.09
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 157.57/57.09
s(mark(z0)) → mark(s(z0)) 157.57/57.09
s(ok(z0)) → ok(s(z0)) 157.57/57.09
cons(mark(z0), z1) → mark(cons(z0, z1)) 157.57/57.09
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 157.57/57.09
from(mark(z0)) → mark(from(z0)) 157.57/57.09
from(ok(z0)) → ok(from(z0)) 157.57/57.09
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 157.57/57.09
proper(0) → ok(0) 157.57/57.09
proper(nil) → ok(nil) 157.57/57.09
proper(s(z0)) → s(proper(z0)) 157.57/57.09
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 157.57/57.09
proper(from(z0)) → from(proper(z0)) 157.57/57.09
top(mark(z0)) → top(proper(z0)) 157.57/57.09
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(first(z0, z1)) → c3(FIRST(active(z0), z1), ACTIVE(z0)) 157.57/57.09
ACTIVE(first(z0, z1)) → c4(FIRST(z0, active(z1)), ACTIVE(z1)) 157.57/57.09
ACTIVE(from(z0)) → c7(FROM(active(z0)), ACTIVE(z0)) 157.57/57.09
FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.57/57.09
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.57/57.09
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.57/57.09
S(mark(z0)) → c11(S(z0)) 157.57/57.09
S(ok(z0)) → c12(S(z0)) 157.57/57.09
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.57/57.09
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.57/57.09
FROM(mark(z0)) → c15(FROM(z0)) 157.57/57.09
FROM(ok(z0)) → c16(FROM(z0)) 157.57/57.09
PROPER(first(z0, z1)) → c17(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 157.57/57.09
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 157.57/57.09
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 157.57/57.09
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 157.57/57.09
TOP(mark(z0)) → c23(TOP(proper(z0)), PROPER(z0)) 157.57/57.09
TOP(ok(z0)) → c24(TOP(active(z0)), ACTIVE(z0)) 157.57/57.09
ACTIVE(from(z0)) → c2(S(z0)) 157.57/57.09
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 157.57/57.09
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 157.57/57.09
ACTIVE(s(first(0, z0))) → c5(S(mark(nil)), ACTIVE(first(0, z0))) 157.57/57.09
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c5(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.57/57.09
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.57/57.09
ACTIVE(s(first(z0, z1))) → c5(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.57/57.09
ACTIVE(s(first(z0, z1))) → c5(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.57/57.09
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 157.57/57.09
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.57/57.09
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 157.57/57.09
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 157.57/57.09
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 157.57/57.09
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 157.57/57.09
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 157.57/57.09
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 157.57/57.09
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 157.57/57.09
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 157.57/57.09
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
S tuples:

ACTIVE(first(z0, z1)) → c3(FIRST(active(z0), z1), ACTIVE(z0)) 157.57/57.09
ACTIVE(first(z0, z1)) → c4(FIRST(z0, active(z1)), ACTIVE(z1)) 157.57/57.09
ACTIVE(from(z0)) → c7(FROM(active(z0)), ACTIVE(z0)) 157.57/57.09
FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.57/57.09
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.57/57.09
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.57/57.09
S(mark(z0)) → c11(S(z0)) 157.57/57.09
S(ok(z0)) → c12(S(z0)) 157.57/57.09
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.57/57.09
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.57/57.09
FROM(mark(z0)) → c15(FROM(z0)) 157.57/57.09
FROM(ok(z0)) → c16(FROM(z0)) 157.57/57.09
PROPER(first(z0, z1)) → c17(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 157.57/57.09
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 157.57/57.09
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 157.57/57.09
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 157.57/57.09
TOP(ok(z0)) → c24(TOP(active(z0)), ACTIVE(z0)) 157.57/57.09
ACTIVE(from(z0)) → c2(S(z0)) 157.57/57.09
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 157.57/57.09
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 157.57/57.09
ACTIVE(s(first(0, z0))) → c5(S(mark(nil)), ACTIVE(first(0, z0))) 157.57/57.09
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c5(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.57/57.09
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.57/57.09
ACTIVE(s(first(z0, z1))) → c5(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.57/57.09
ACTIVE(s(first(z0, z1))) → c5(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.57/57.09
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 157.57/57.09
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.14
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 157.90/57.14
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 157.90/57.14
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.14
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 157.90/57.14
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 157.90/57.14
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 157.90/57.14
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 157.90/57.14
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 157.90/57.14
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
K tuples:

TOP(mark(z0)) → c23(TOP(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, first, s, cons, from, proper, top

Defined Pair Symbols:

ACTIVE, FIRST, S, CONS, FROM, PROPER, TOP

Compound Symbols:

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

157.90/57.14
157.90/57.14

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

Use narrowing to replace ACTIVE(from(z0)) → c7(FROM(active(z0)), ACTIVE(z0)) by

ACTIVE(from(first(0, z0))) → c7(FROM(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.14
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c7(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.14
ACTIVE(from(from(z0))) → c7(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.14
ACTIVE(from(first(z0, z1))) → c7(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.14
ACTIVE(from(first(z0, z1))) → c7(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.14
ACTIVE(from(s(z0))) → c7(FROM(s(active(z0))), ACTIVE(s(z0))) 157.90/57.14
ACTIVE(from(cons(z0, z1))) → c7(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.14
ACTIVE(from(from(z0))) → c7(FROM(from(active(z0))), ACTIVE(from(z0)))
157.90/57.14
157.90/57.14

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil) 157.90/57.14
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 157.90/57.14
active(from(z0)) → mark(cons(z0, from(s(z0)))) 157.90/57.14
active(first(z0, z1)) → first(active(z0), z1) 157.90/57.14
active(first(z0, z1)) → first(z0, active(z1)) 157.90/57.14
active(s(z0)) → s(active(z0)) 157.90/57.14
active(cons(z0, z1)) → cons(active(z0), z1) 157.90/57.14
active(from(z0)) → from(active(z0)) 157.90/57.14
first(mark(z0), z1) → mark(first(z0, z1)) 157.90/57.14
first(z0, mark(z1)) → mark(first(z0, z1)) 157.90/57.14
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 157.90/57.14
s(mark(z0)) → mark(s(z0)) 157.90/57.14
s(ok(z0)) → ok(s(z0)) 157.90/57.14
cons(mark(z0), z1) → mark(cons(z0, z1)) 157.90/57.14
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 157.90/57.14
from(mark(z0)) → mark(from(z0)) 157.90/57.14
from(ok(z0)) → ok(from(z0)) 157.90/57.14
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 157.90/57.14
proper(0) → ok(0) 157.90/57.14
proper(nil) → ok(nil) 157.90/57.14
proper(s(z0)) → s(proper(z0)) 157.90/57.14
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 157.90/57.14
proper(from(z0)) → from(proper(z0)) 157.90/57.14
top(mark(z0)) → top(proper(z0)) 157.90/57.14
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(first(z0, z1)) → c3(FIRST(active(z0), z1), ACTIVE(z0)) 157.90/57.14
ACTIVE(first(z0, z1)) → c4(FIRST(z0, active(z1)), ACTIVE(z1)) 157.90/57.14
FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.90/57.14
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.90/57.14
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.90/57.14
S(mark(z0)) → c11(S(z0)) 157.90/57.14
S(ok(z0)) → c12(S(z0)) 157.90/57.14
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.90/57.14
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.90/57.14
FROM(mark(z0)) → c15(FROM(z0)) 157.90/57.14
FROM(ok(z0)) → c16(FROM(z0)) 157.90/57.14
PROPER(first(z0, z1)) → c17(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 157.90/57.14
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 157.90/57.14
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 157.90/57.14
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 157.90/57.14
TOP(mark(z0)) → c23(TOP(proper(z0)), PROPER(z0)) 157.90/57.14
TOP(ok(z0)) → c24(TOP(active(z0)), ACTIVE(z0)) 157.90/57.14
ACTIVE(from(z0)) → c2(S(z0)) 157.90/57.14
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 157.90/57.14
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 157.90/57.14
ACTIVE(s(first(0, z0))) → c5(S(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.14
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c5(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.14
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.14
ACTIVE(s(first(z0, z1))) → c5(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.14
ACTIVE(s(first(z0, z1))) → c5(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.14
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 157.90/57.14
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.14
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 157.90/57.14
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 157.90/57.14
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.14
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 157.90/57.14
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 157.90/57.14
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 157.90/57.14
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 157.90/57.14
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 157.90/57.14
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 157.90/57.14
ACTIVE(from(first(0, z0))) → c7(FROM(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.14
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c7(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.14
ACTIVE(from(from(z0))) → c7(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.14
ACTIVE(from(first(z0, z1))) → c7(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.14
ACTIVE(from(first(z0, z1))) → c7(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.14
ACTIVE(from(s(z0))) → c7(FROM(s(active(z0))), ACTIVE(s(z0))) 157.90/57.14
ACTIVE(from(cons(z0, z1))) → c7(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.14
ACTIVE(from(from(z0))) → c7(FROM(from(active(z0))), ACTIVE(from(z0)))
S tuples:

ACTIVE(first(z0, z1)) → c3(FIRST(active(z0), z1), ACTIVE(z0)) 157.90/57.14
ACTIVE(first(z0, z1)) → c4(FIRST(z0, active(z1)), ACTIVE(z1)) 157.90/57.14
FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.90/57.14
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.90/57.14
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.90/57.14
S(mark(z0)) → c11(S(z0)) 157.90/57.14
S(ok(z0)) → c12(S(z0)) 157.90/57.14
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.90/57.14
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.90/57.14
FROM(mark(z0)) → c15(FROM(z0)) 157.90/57.14
FROM(ok(z0)) → c16(FROM(z0)) 157.90/57.14
PROPER(first(z0, z1)) → c17(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 157.90/57.14
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 157.90/57.14
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 157.90/57.14
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 157.90/57.14
TOP(ok(z0)) → c24(TOP(active(z0)), ACTIVE(z0)) 157.90/57.14
ACTIVE(from(z0)) → c2(S(z0)) 157.90/57.14
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 157.90/57.14
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 157.90/57.14
ACTIVE(s(first(0, z0))) → c5(S(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.14
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c5(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.14
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.14
ACTIVE(s(first(z0, z1))) → c5(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.14
ACTIVE(s(first(z0, z1))) → c5(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.14
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 157.90/57.14
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.14
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 157.90/57.14
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 157.90/57.14
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.14
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 157.90/57.14
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 157.90/57.14
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 157.90/57.14
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 157.90/57.14
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 157.90/57.14
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 157.90/57.14
ACTIVE(from(first(0, z0))) → c7(FROM(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.14
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c7(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.14
ACTIVE(from(from(z0))) → c7(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.14
ACTIVE(from(first(z0, z1))) → c7(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.14
ACTIVE(from(first(z0, z1))) → c7(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.14
ACTIVE(from(s(z0))) → c7(FROM(s(active(z0))), ACTIVE(s(z0))) 157.90/57.14
ACTIVE(from(cons(z0, z1))) → c7(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.14
ACTIVE(from(from(z0))) → c7(FROM(from(active(z0))), ACTIVE(from(z0)))
K tuples:

TOP(mark(z0)) → c23(TOP(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, first, s, cons, from, proper, top

Defined Pair Symbols:

ACTIVE, FIRST, S, CONS, FROM, PROPER, TOP

Compound Symbols:

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

157.90/57.16
157.90/57.16

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

Use narrowing to replace PROPER(first(z0, z1)) → c17(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(first(x0, first(z0, z1))) → c17(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 157.90/57.16
PROPER(first(x0, 0)) → c17(FIRST(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 157.90/57.16
PROPER(first(x0, nil)) → c17(FIRST(proper(x0), ok(nil)), PROPER(x0), PROPER(nil)) 157.90/57.16
PROPER(first(x0, s(z0))) → c17(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 157.90/57.16
PROPER(first(x0, cons(z0, z1))) → c17(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 157.90/57.16
PROPER(first(x0, from(z0))) → c17(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 157.90/57.16
PROPER(first(first(z0, z1), x1)) → c17(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 157.90/57.16
PROPER(first(0, x1)) → c17(FIRST(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 157.90/57.16
PROPER(first(nil, x1)) → c17(FIRST(ok(nil), proper(x1)), PROPER(nil), PROPER(x1)) 157.90/57.16
PROPER(first(s(z0), x1)) → c17(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 157.90/57.16
PROPER(first(cons(z0, z1), x1)) → c17(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 157.90/57.16
PROPER(first(from(z0), x1)) → c17(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
157.90/57.16
157.90/57.16

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil) 157.90/57.16
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 157.90/57.16
active(from(z0)) → mark(cons(z0, from(s(z0)))) 157.90/57.16
active(first(z0, z1)) → first(active(z0), z1) 157.90/57.16
active(first(z0, z1)) → first(z0, active(z1)) 157.90/57.16
active(s(z0)) → s(active(z0)) 157.90/57.16
active(cons(z0, z1)) → cons(active(z0), z1) 157.90/57.16
active(from(z0)) → from(active(z0)) 157.90/57.16
first(mark(z0), z1) → mark(first(z0, z1)) 157.90/57.16
first(z0, mark(z1)) → mark(first(z0, z1)) 157.90/57.16
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 157.90/57.16
s(mark(z0)) → mark(s(z0)) 157.90/57.16
s(ok(z0)) → ok(s(z0)) 157.90/57.16
cons(mark(z0), z1) → mark(cons(z0, z1)) 157.90/57.16
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 157.90/57.16
from(mark(z0)) → mark(from(z0)) 157.90/57.16
from(ok(z0)) → ok(from(z0)) 157.90/57.16
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 157.90/57.16
proper(0) → ok(0) 157.90/57.16
proper(nil) → ok(nil) 157.90/57.16
proper(s(z0)) → s(proper(z0)) 157.90/57.16
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 157.90/57.16
proper(from(z0)) → from(proper(z0)) 157.90/57.16
top(mark(z0)) → top(proper(z0)) 157.90/57.16
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(first(z0, z1)) → c3(FIRST(active(z0), z1), ACTIVE(z0)) 157.90/57.16
ACTIVE(first(z0, z1)) → c4(FIRST(z0, active(z1)), ACTIVE(z1)) 157.90/57.16
FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.90/57.16
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.90/57.16
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.90/57.16
S(mark(z0)) → c11(S(z0)) 157.90/57.16
S(ok(z0)) → c12(S(z0)) 157.90/57.16
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.90/57.16
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.90/57.16
FROM(mark(z0)) → c15(FROM(z0)) 157.90/57.16
FROM(ok(z0)) → c16(FROM(z0)) 157.90/57.16
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 157.90/57.16
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 157.90/57.16
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 157.90/57.16
TOP(mark(z0)) → c23(TOP(proper(z0)), PROPER(z0)) 157.90/57.16
TOP(ok(z0)) → c24(TOP(active(z0)), ACTIVE(z0)) 157.90/57.16
ACTIVE(from(z0)) → c2(S(z0)) 157.90/57.16
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 157.90/57.16
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 157.90/57.16
ACTIVE(s(first(0, z0))) → c5(S(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.16
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c5(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.16
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.16
ACTIVE(s(first(z0, z1))) → c5(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.16
ACTIVE(s(first(z0, z1))) → c5(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.16
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 157.90/57.16
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.16
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 157.90/57.16
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 157.90/57.16
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.16
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 157.90/57.16
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 157.90/57.16
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 157.90/57.16
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 157.90/57.16
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 157.90/57.16
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 157.90/57.16
ACTIVE(from(first(0, z0))) → c7(FROM(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.16
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c7(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.16
ACTIVE(from(from(z0))) → c7(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.16
ACTIVE(from(first(z0, z1))) → c7(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.16
ACTIVE(from(first(z0, z1))) → c7(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.16
ACTIVE(from(s(z0))) → c7(FROM(s(active(z0))), ACTIVE(s(z0))) 157.90/57.16
ACTIVE(from(cons(z0, z1))) → c7(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.16
ACTIVE(from(from(z0))) → c7(FROM(from(active(z0))), ACTIVE(from(z0))) 157.90/57.16
PROPER(first(x0, first(z0, z1))) → c17(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 157.90/57.16
PROPER(first(x0, 0)) → c17(FIRST(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 157.90/57.16
PROPER(first(x0, nil)) → c17(FIRST(proper(x0), ok(nil)), PROPER(x0), PROPER(nil)) 157.90/57.16
PROPER(first(x0, s(z0))) → c17(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 157.90/57.16
PROPER(first(x0, cons(z0, z1))) → c17(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 157.90/57.16
PROPER(first(x0, from(z0))) → c17(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 157.90/57.16
PROPER(first(first(z0, z1), x1)) → c17(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 157.90/57.16
PROPER(first(0, x1)) → c17(FIRST(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 157.90/57.17
PROPER(first(nil, x1)) → c17(FIRST(ok(nil), proper(x1)), PROPER(nil), PROPER(x1)) 157.90/57.17
PROPER(first(s(z0), x1)) → c17(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 157.90/57.17
PROPER(first(cons(z0, z1), x1)) → c17(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(first(from(z0), x1)) → c17(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
S tuples:

ACTIVE(first(z0, z1)) → c3(FIRST(active(z0), z1), ACTIVE(z0)) 157.90/57.17
ACTIVE(first(z0, z1)) → c4(FIRST(z0, active(z1)), ACTIVE(z1)) 157.90/57.17
FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.90/57.17
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.90/57.17
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.90/57.17
S(mark(z0)) → c11(S(z0)) 157.90/57.17
S(ok(z0)) → c12(S(z0)) 157.90/57.17
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.90/57.17
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.90/57.17
FROM(mark(z0)) → c15(FROM(z0)) 157.90/57.17
FROM(ok(z0)) → c16(FROM(z0)) 157.90/57.17
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 157.90/57.17
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 157.90/57.17
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 157.90/57.17
TOP(ok(z0)) → c24(TOP(active(z0)), ACTIVE(z0)) 157.90/57.17
ACTIVE(from(z0)) → c2(S(z0)) 157.90/57.17
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 157.90/57.17
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 157.90/57.17
ACTIVE(s(first(0, z0))) → c5(S(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c5(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(s(first(z0, z1))) → c5(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(s(first(z0, z1))) → c5(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(from(first(0, z0))) → c7(FROM(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c7(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(from(from(z0))) → c7(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(from(first(z0, z1))) → c7(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(from(first(z0, z1))) → c7(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(from(s(z0))) → c7(FROM(s(active(z0))), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(from(cons(z0, z1))) → c7(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(from(from(z0))) → c7(FROM(from(active(z0))), ACTIVE(from(z0))) 157.90/57.17
PROPER(first(x0, first(z0, z1))) → c17(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 157.90/57.17
PROPER(first(x0, 0)) → c17(FIRST(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 157.90/57.17
PROPER(first(x0, nil)) → c17(FIRST(proper(x0), ok(nil)), PROPER(x0), PROPER(nil)) 157.90/57.17
PROPER(first(x0, s(z0))) → c17(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 157.90/57.17
PROPER(first(x0, cons(z0, z1))) → c17(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 157.90/57.17
PROPER(first(x0, from(z0))) → c17(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 157.90/57.17
PROPER(first(first(z0, z1), x1)) → c17(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(first(0, x1)) → c17(FIRST(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 157.90/57.17
PROPER(first(nil, x1)) → c17(FIRST(ok(nil), proper(x1)), PROPER(nil), PROPER(x1)) 157.90/57.17
PROPER(first(s(z0), x1)) → c17(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 157.90/57.17
PROPER(first(cons(z0, z1), x1)) → c17(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(first(from(z0), x1)) → c17(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
K tuples:

TOP(mark(z0)) → c23(TOP(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, first, s, cons, from, proper, top

Defined Pair Symbols:

ACTIVE, FIRST, S, CONS, FROM, PROPER, TOP

Compound Symbols:

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

157.90/57.17
157.90/57.17

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

Removed 4 trailing tuple parts
157.90/57.17
157.90/57.17

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil) 157.90/57.17
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 157.90/57.17
active(from(z0)) → mark(cons(z0, from(s(z0)))) 157.90/57.17
active(first(z0, z1)) → first(active(z0), z1) 157.90/57.17
active(first(z0, z1)) → first(z0, active(z1)) 157.90/57.17
active(s(z0)) → s(active(z0)) 157.90/57.17
active(cons(z0, z1)) → cons(active(z0), z1) 157.90/57.17
active(from(z0)) → from(active(z0)) 157.90/57.17
first(mark(z0), z1) → mark(first(z0, z1)) 157.90/57.17
first(z0, mark(z1)) → mark(first(z0, z1)) 157.90/57.17
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 157.90/57.17
s(mark(z0)) → mark(s(z0)) 157.90/57.17
s(ok(z0)) → ok(s(z0)) 157.90/57.17
cons(mark(z0), z1) → mark(cons(z0, z1)) 157.90/57.17
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 157.90/57.17
from(mark(z0)) → mark(from(z0)) 157.90/57.17
from(ok(z0)) → ok(from(z0)) 157.90/57.17
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 157.90/57.17
proper(0) → ok(0) 157.90/57.17
proper(nil) → ok(nil) 157.90/57.17
proper(s(z0)) → s(proper(z0)) 157.90/57.17
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 157.90/57.17
proper(from(z0)) → from(proper(z0)) 157.90/57.17
top(mark(z0)) → top(proper(z0)) 157.90/57.17
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(first(z0, z1)) → c3(FIRST(active(z0), z1), ACTIVE(z0)) 157.90/57.17
ACTIVE(first(z0, z1)) → c4(FIRST(z0, active(z1)), ACTIVE(z1)) 157.90/57.17
FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.90/57.17
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.90/57.17
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.90/57.17
S(mark(z0)) → c11(S(z0)) 157.90/57.17
S(ok(z0)) → c12(S(z0)) 157.90/57.17
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.90/57.17
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.90/57.17
FROM(mark(z0)) → c15(FROM(z0)) 157.90/57.17
FROM(ok(z0)) → c16(FROM(z0)) 157.90/57.17
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 157.90/57.17
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 157.90/57.17
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 157.90/57.17
TOP(mark(z0)) → c23(TOP(proper(z0)), PROPER(z0)) 157.90/57.17
TOP(ok(z0)) → c24(TOP(active(z0)), ACTIVE(z0)) 157.90/57.17
ACTIVE(from(z0)) → c2(S(z0)) 157.90/57.17
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 157.90/57.17
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 157.90/57.17
ACTIVE(s(first(0, z0))) → c5(S(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c5(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(s(first(z0, z1))) → c5(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(s(first(z0, z1))) → c5(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(from(first(0, z0))) → c7(FROM(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c7(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(from(from(z0))) → c7(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(from(first(z0, z1))) → c7(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(from(first(z0, z1))) → c7(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(from(s(z0))) → c7(FROM(s(active(z0))), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(from(cons(z0, z1))) → c7(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(from(from(z0))) → c7(FROM(from(active(z0))), ACTIVE(from(z0))) 157.90/57.17
PROPER(first(x0, first(z0, z1))) → c17(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 157.90/57.17
PROPER(first(x0, s(z0))) → c17(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 157.90/57.17
PROPER(first(x0, cons(z0, z1))) → c17(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 157.90/57.17
PROPER(first(x0, from(z0))) → c17(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 157.90/57.17
PROPER(first(first(z0, z1), x1)) → c17(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(first(s(z0), x1)) → c17(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 157.90/57.17
PROPER(first(cons(z0, z1), x1)) → c17(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(first(from(z0), x1)) → c17(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 157.90/57.17
PROPER(first(x0, 0)) → c17(FIRST(proper(x0), ok(0)), PROPER(x0)) 157.90/57.17
PROPER(first(x0, nil)) → c17(FIRST(proper(x0), ok(nil)), PROPER(x0)) 157.90/57.17
PROPER(first(0, x1)) → c17(FIRST(ok(0), proper(x1)), PROPER(x1)) 157.90/57.17
PROPER(first(nil, x1)) → c17(FIRST(ok(nil), proper(x1)), PROPER(x1))
S tuples:

ACTIVE(first(z0, z1)) → c3(FIRST(active(z0), z1), ACTIVE(z0)) 157.90/57.17
ACTIVE(first(z0, z1)) → c4(FIRST(z0, active(z1)), ACTIVE(z1)) 157.90/57.17
FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.90/57.17
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.90/57.17
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.90/57.17
S(mark(z0)) → c11(S(z0)) 157.90/57.17
S(ok(z0)) → c12(S(z0)) 157.90/57.17
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.90/57.17
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.90/57.17
FROM(mark(z0)) → c15(FROM(z0)) 157.90/57.17
FROM(ok(z0)) → c16(FROM(z0)) 157.90/57.17
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 157.90/57.17
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 157.90/57.17
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 157.90/57.17
TOP(ok(z0)) → c24(TOP(active(z0)), ACTIVE(z0)) 157.90/57.17
ACTIVE(from(z0)) → c2(S(z0)) 157.90/57.17
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 157.90/57.17
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 157.90/57.17
ACTIVE(s(first(0, z0))) → c5(S(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c5(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(s(first(z0, z1))) → c5(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(s(first(z0, z1))) → c5(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(from(first(0, z0))) → c7(FROM(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c7(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(from(from(z0))) → c7(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(from(first(z0, z1))) → c7(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(from(first(z0, z1))) → c7(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(from(s(z0))) → c7(FROM(s(active(z0))), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(from(cons(z0, z1))) → c7(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(from(from(z0))) → c7(FROM(from(active(z0))), ACTIVE(from(z0))) 157.90/57.17
PROPER(first(x0, first(z0, z1))) → c17(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 157.90/57.17
PROPER(first(x0, s(z0))) → c17(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 157.90/57.17
PROPER(first(x0, cons(z0, z1))) → c17(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 157.90/57.17
PROPER(first(x0, from(z0))) → c17(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 157.90/57.17
PROPER(first(first(z0, z1), x1)) → c17(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(first(s(z0), x1)) → c17(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 157.90/57.17
PROPER(first(cons(z0, z1), x1)) → c17(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(first(from(z0), x1)) → c17(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 157.90/57.17
PROPER(first(x0, 0)) → c17(FIRST(proper(x0), ok(0)), PROPER(x0)) 157.90/57.17
PROPER(first(x0, nil)) → c17(FIRST(proper(x0), ok(nil)), PROPER(x0)) 157.90/57.17
PROPER(first(0, x1)) → c17(FIRST(ok(0), proper(x1)), PROPER(x1)) 157.90/57.17
PROPER(first(nil, x1)) → c17(FIRST(ok(nil), proper(x1)), PROPER(x1))
K tuples:

TOP(mark(z0)) → c23(TOP(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, first, s, cons, from, proper, top

Defined Pair Symbols:

ACTIVE, FIRST, S, CONS, FROM, PROPER, TOP

Compound Symbols:

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

157.90/57.17
157.90/57.17

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

Use narrowing to replace PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) by

PROPER(s(first(z0, z1))) → c20(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 157.90/57.17
PROPER(s(0)) → c20(S(ok(0)), PROPER(0)) 157.90/57.17
PROPER(s(nil)) → c20(S(ok(nil)), PROPER(nil)) 157.90/57.17
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 157.90/57.17
PROPER(s(cons(z0, z1))) → c20(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 157.90/57.17
PROPER(s(from(z0))) → c20(S(from(proper(z0))), PROPER(from(z0)))
157.90/57.17
157.90/57.17

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil) 157.90/57.17
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 157.90/57.17
active(from(z0)) → mark(cons(z0, from(s(z0)))) 157.90/57.17
active(first(z0, z1)) → first(active(z0), z1) 157.90/57.17
active(first(z0, z1)) → first(z0, active(z1)) 157.90/57.17
active(s(z0)) → s(active(z0)) 157.90/57.17
active(cons(z0, z1)) → cons(active(z0), z1) 157.90/57.17
active(from(z0)) → from(active(z0)) 157.90/57.17
first(mark(z0), z1) → mark(first(z0, z1)) 157.90/57.17
first(z0, mark(z1)) → mark(first(z0, z1)) 157.90/57.17
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 157.90/57.17
s(mark(z0)) → mark(s(z0)) 157.90/57.17
s(ok(z0)) → ok(s(z0)) 157.90/57.17
cons(mark(z0), z1) → mark(cons(z0, z1)) 157.90/57.17
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 157.90/57.17
from(mark(z0)) → mark(from(z0)) 157.90/57.17
from(ok(z0)) → ok(from(z0)) 157.90/57.17
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 157.90/57.17
proper(0) → ok(0) 157.90/57.17
proper(nil) → ok(nil) 157.90/57.17
proper(s(z0)) → s(proper(z0)) 157.90/57.17
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 157.90/57.17
proper(from(z0)) → from(proper(z0)) 157.90/57.17
top(mark(z0)) → top(proper(z0)) 157.90/57.17
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(first(z0, z1)) → c3(FIRST(active(z0), z1), ACTIVE(z0)) 157.90/57.17
ACTIVE(first(z0, z1)) → c4(FIRST(z0, active(z1)), ACTIVE(z1)) 157.90/57.17
FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.90/57.17
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.90/57.17
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.90/57.17
S(mark(z0)) → c11(S(z0)) 157.90/57.17
S(ok(z0)) → c12(S(z0)) 157.90/57.17
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.90/57.17
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.90/57.17
FROM(mark(z0)) → c15(FROM(z0)) 157.90/57.17
FROM(ok(z0)) → c16(FROM(z0)) 157.90/57.17
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 157.90/57.17
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 157.90/57.17
TOP(mark(z0)) → c23(TOP(proper(z0)), PROPER(z0)) 157.90/57.17
TOP(ok(z0)) → c24(TOP(active(z0)), ACTIVE(z0)) 157.90/57.17
ACTIVE(from(z0)) → c2(S(z0)) 157.90/57.17
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 157.90/57.17
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 157.90/57.17
ACTIVE(s(first(0, z0))) → c5(S(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c5(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(s(first(z0, z1))) → c5(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(s(first(z0, z1))) → c5(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(from(first(0, z0))) → c7(FROM(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c7(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(from(from(z0))) → c7(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(from(first(z0, z1))) → c7(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(from(first(z0, z1))) → c7(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(from(s(z0))) → c7(FROM(s(active(z0))), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(from(cons(z0, z1))) → c7(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(from(from(z0))) → c7(FROM(from(active(z0))), ACTIVE(from(z0))) 157.90/57.17
PROPER(first(x0, first(z0, z1))) → c17(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 157.90/57.17
PROPER(first(x0, s(z0))) → c17(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 157.90/57.17
PROPER(first(x0, cons(z0, z1))) → c17(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 157.90/57.17
PROPER(first(x0, from(z0))) → c17(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 157.90/57.17
PROPER(first(first(z0, z1), x1)) → c17(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(first(s(z0), x1)) → c17(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 157.90/57.17
PROPER(first(cons(z0, z1), x1)) → c17(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(first(from(z0), x1)) → c17(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 157.90/57.17
PROPER(first(x0, 0)) → c17(FIRST(proper(x0), ok(0)), PROPER(x0)) 157.90/57.17
PROPER(first(x0, nil)) → c17(FIRST(proper(x0), ok(nil)), PROPER(x0)) 157.90/57.17
PROPER(first(0, x1)) → c17(FIRST(ok(0), proper(x1)), PROPER(x1)) 157.90/57.17
PROPER(first(nil, x1)) → c17(FIRST(ok(nil), proper(x1)), PROPER(x1)) 157.90/57.17
PROPER(s(first(z0, z1))) → c20(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 157.90/57.17
PROPER(s(0)) → c20(S(ok(0)), PROPER(0)) 157.90/57.17
PROPER(s(nil)) → c20(S(ok(nil)), PROPER(nil)) 157.90/57.17
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 157.90/57.17
PROPER(s(cons(z0, z1))) → c20(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 157.90/57.17
PROPER(s(from(z0))) → c20(S(from(proper(z0))), PROPER(from(z0)))
S tuples:

ACTIVE(first(z0, z1)) → c3(FIRST(active(z0), z1), ACTIVE(z0)) 157.90/57.17
ACTIVE(first(z0, z1)) → c4(FIRST(z0, active(z1)), ACTIVE(z1)) 157.90/57.17
FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.90/57.17
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.90/57.17
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.90/57.17
S(mark(z0)) → c11(S(z0)) 157.90/57.17
S(ok(z0)) → c12(S(z0)) 157.90/57.17
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.90/57.17
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.90/57.17
FROM(mark(z0)) → c15(FROM(z0)) 157.90/57.17
FROM(ok(z0)) → c16(FROM(z0)) 157.90/57.17
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 157.90/57.17
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 157.90/57.17
TOP(ok(z0)) → c24(TOP(active(z0)), ACTIVE(z0)) 157.90/57.17
ACTIVE(from(z0)) → c2(S(z0)) 157.90/57.17
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 157.90/57.17
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 157.90/57.17
ACTIVE(s(first(0, z0))) → c5(S(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c5(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(s(first(z0, z1))) → c5(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(s(first(z0, z1))) → c5(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(from(first(0, z0))) → c7(FROM(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c7(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(from(from(z0))) → c7(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(from(first(z0, z1))) → c7(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(from(first(z0, z1))) → c7(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(from(s(z0))) → c7(FROM(s(active(z0))), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(from(cons(z0, z1))) → c7(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(from(from(z0))) → c7(FROM(from(active(z0))), ACTIVE(from(z0))) 157.90/57.17
PROPER(first(x0, first(z0, z1))) → c17(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 157.90/57.17
PROPER(first(x0, s(z0))) → c17(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 157.90/57.17
PROPER(first(x0, cons(z0, z1))) → c17(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 157.90/57.17
PROPER(first(x0, from(z0))) → c17(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 157.90/57.17
PROPER(first(first(z0, z1), x1)) → c17(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(first(s(z0), x1)) → c17(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 157.90/57.17
PROPER(first(cons(z0, z1), x1)) → c17(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(first(from(z0), x1)) → c17(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 157.90/57.17
PROPER(first(x0, 0)) → c17(FIRST(proper(x0), ok(0)), PROPER(x0)) 157.90/57.17
PROPER(first(x0, nil)) → c17(FIRST(proper(x0), ok(nil)), PROPER(x0)) 157.90/57.17
PROPER(first(0, x1)) → c17(FIRST(ok(0), proper(x1)), PROPER(x1)) 157.90/57.17
PROPER(first(nil, x1)) → c17(FIRST(ok(nil), proper(x1)), PROPER(x1)) 157.90/57.17
PROPER(s(first(z0, z1))) → c20(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 157.90/57.17
PROPER(s(0)) → c20(S(ok(0)), PROPER(0)) 157.90/57.17
PROPER(s(nil)) → c20(S(ok(nil)), PROPER(nil)) 157.90/57.17
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 157.90/57.17
PROPER(s(cons(z0, z1))) → c20(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 157.90/57.17
PROPER(s(from(z0))) → c20(S(from(proper(z0))), PROPER(from(z0)))
K tuples:

TOP(mark(z0)) → c23(TOP(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, first, s, cons, from, proper, top

Defined Pair Symbols:

ACTIVE, FIRST, S, CONS, FROM, PROPER, TOP

Compound Symbols:

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

157.90/57.17
157.90/57.17

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

Removed 2 trailing tuple parts
157.90/57.17
157.90/57.17

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil) 157.90/57.17
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 157.90/57.17
active(from(z0)) → mark(cons(z0, from(s(z0)))) 157.90/57.17
active(first(z0, z1)) → first(active(z0), z1) 157.90/57.17
active(first(z0, z1)) → first(z0, active(z1)) 157.90/57.17
active(s(z0)) → s(active(z0)) 157.90/57.17
active(cons(z0, z1)) → cons(active(z0), z1) 157.90/57.17
active(from(z0)) → from(active(z0)) 157.90/57.17
first(mark(z0), z1) → mark(first(z0, z1)) 157.90/57.17
first(z0, mark(z1)) → mark(first(z0, z1)) 157.90/57.17
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 157.90/57.17
s(mark(z0)) → mark(s(z0)) 157.90/57.17
s(ok(z0)) → ok(s(z0)) 157.90/57.17
cons(mark(z0), z1) → mark(cons(z0, z1)) 157.90/57.17
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 157.90/57.17
from(mark(z0)) → mark(from(z0)) 157.90/57.17
from(ok(z0)) → ok(from(z0)) 157.90/57.17
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 157.90/57.17
proper(0) → ok(0) 157.90/57.17
proper(nil) → ok(nil) 157.90/57.17
proper(s(z0)) → s(proper(z0)) 157.90/57.17
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 157.90/57.17
proper(from(z0)) → from(proper(z0)) 157.90/57.17
top(mark(z0)) → top(proper(z0)) 157.90/57.17
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(first(z0, z1)) → c3(FIRST(active(z0), z1), ACTIVE(z0)) 157.90/57.17
ACTIVE(first(z0, z1)) → c4(FIRST(z0, active(z1)), ACTIVE(z1)) 157.90/57.17
FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.90/57.17
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.90/57.17
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.90/57.17
S(mark(z0)) → c11(S(z0)) 157.90/57.17
S(ok(z0)) → c12(S(z0)) 157.90/57.17
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.90/57.17
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.90/57.17
FROM(mark(z0)) → c15(FROM(z0)) 157.90/57.17
FROM(ok(z0)) → c16(FROM(z0)) 157.90/57.17
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 157.90/57.17
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 157.90/57.17
TOP(mark(z0)) → c23(TOP(proper(z0)), PROPER(z0)) 157.90/57.17
TOP(ok(z0)) → c24(TOP(active(z0)), ACTIVE(z0)) 157.90/57.17
ACTIVE(from(z0)) → c2(S(z0)) 157.90/57.17
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 157.90/57.17
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 157.90/57.17
ACTIVE(s(first(0, z0))) → c5(S(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c5(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(s(first(z0, z1))) → c5(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(s(first(z0, z1))) → c5(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(from(first(0, z0))) → c7(FROM(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c7(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(from(from(z0))) → c7(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(from(first(z0, z1))) → c7(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(from(first(z0, z1))) → c7(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(from(s(z0))) → c7(FROM(s(active(z0))), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(from(cons(z0, z1))) → c7(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(from(from(z0))) → c7(FROM(from(active(z0))), ACTIVE(from(z0))) 157.90/57.17
PROPER(first(x0, first(z0, z1))) → c17(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 157.90/57.17
PROPER(first(x0, s(z0))) → c17(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 157.90/57.17
PROPER(first(x0, cons(z0, z1))) → c17(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 157.90/57.17
PROPER(first(x0, from(z0))) → c17(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 157.90/57.17
PROPER(first(first(z0, z1), x1)) → c17(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(first(s(z0), x1)) → c17(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 157.90/57.17
PROPER(first(cons(z0, z1), x1)) → c17(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(first(from(z0), x1)) → c17(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 157.90/57.17
PROPER(first(x0, 0)) → c17(FIRST(proper(x0), ok(0)), PROPER(x0)) 157.90/57.17
PROPER(first(x0, nil)) → c17(FIRST(proper(x0), ok(nil)), PROPER(x0)) 157.90/57.17
PROPER(first(0, x1)) → c17(FIRST(ok(0), proper(x1)), PROPER(x1)) 157.90/57.17
PROPER(first(nil, x1)) → c17(FIRST(ok(nil), proper(x1)), PROPER(x1)) 157.90/57.17
PROPER(s(first(z0, z1))) → c20(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 157.90/57.17
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 157.90/57.17
PROPER(s(cons(z0, z1))) → c20(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 157.90/57.17
PROPER(s(from(z0))) → c20(S(from(proper(z0))), PROPER(from(z0))) 157.90/57.17
PROPER(s(0)) → c20(S(ok(0))) 157.90/57.17
PROPER(s(nil)) → c20(S(ok(nil)))
S tuples:

ACTIVE(first(z0, z1)) → c3(FIRST(active(z0), z1), ACTIVE(z0)) 157.90/57.17
ACTIVE(first(z0, z1)) → c4(FIRST(z0, active(z1)), ACTIVE(z1)) 157.90/57.17
FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.90/57.17
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.90/57.17
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.90/57.17
S(mark(z0)) → c11(S(z0)) 157.90/57.17
S(ok(z0)) → c12(S(z0)) 157.90/57.17
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.90/57.17
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.90/57.17
FROM(mark(z0)) → c15(FROM(z0)) 157.90/57.17
FROM(ok(z0)) → c16(FROM(z0)) 157.90/57.17
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 157.90/57.17
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 157.90/57.17
TOP(ok(z0)) → c24(TOP(active(z0)), ACTIVE(z0)) 157.90/57.17
ACTIVE(from(z0)) → c2(S(z0)) 157.90/57.17
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 157.90/57.17
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 157.90/57.17
ACTIVE(s(first(0, z0))) → c5(S(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c5(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(s(first(z0, z1))) → c5(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(s(first(z0, z1))) → c5(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(from(first(0, z0))) → c7(FROM(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c7(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(from(from(z0))) → c7(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(from(first(z0, z1))) → c7(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(from(first(z0, z1))) → c7(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(from(s(z0))) → c7(FROM(s(active(z0))), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(from(cons(z0, z1))) → c7(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(from(from(z0))) → c7(FROM(from(active(z0))), ACTIVE(from(z0))) 157.90/57.17
PROPER(first(x0, first(z0, z1))) → c17(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 157.90/57.17
PROPER(first(x0, s(z0))) → c17(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 157.90/57.17
PROPER(first(x0, cons(z0, z1))) → c17(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 157.90/57.17
PROPER(first(x0, from(z0))) → c17(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 157.90/57.17
PROPER(first(first(z0, z1), x1)) → c17(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(first(s(z0), x1)) → c17(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 157.90/57.17
PROPER(first(cons(z0, z1), x1)) → c17(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(first(from(z0), x1)) → c17(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 157.90/57.17
PROPER(first(x0, 0)) → c17(FIRST(proper(x0), ok(0)), PROPER(x0)) 157.90/57.17
PROPER(first(x0, nil)) → c17(FIRST(proper(x0), ok(nil)), PROPER(x0)) 157.90/57.17
PROPER(first(0, x1)) → c17(FIRST(ok(0), proper(x1)), PROPER(x1)) 157.90/57.17
PROPER(first(nil, x1)) → c17(FIRST(ok(nil), proper(x1)), PROPER(x1)) 157.90/57.17
PROPER(s(first(z0, z1))) → c20(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 157.90/57.17
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 157.90/57.17
PROPER(s(cons(z0, z1))) → c20(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 157.90/57.17
PROPER(s(from(z0))) → c20(S(from(proper(z0))), PROPER(from(z0))) 157.90/57.17
PROPER(s(0)) → c20(S(ok(0))) 157.90/57.17
PROPER(s(nil)) → c20(S(ok(nil)))
K tuples:

TOP(mark(z0)) → c23(TOP(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, first, s, cons, from, proper, top

Defined Pair Symbols:

ACTIVE, FIRST, S, CONS, FROM, PROPER, TOP

Compound Symbols:

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

157.90/57.17
157.90/57.17

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

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

PROPER(cons(x0, first(z0, z1))) → c21(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 157.90/57.17
PROPER(cons(x0, 0)) → c21(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 157.90/57.17
PROPER(cons(x0, nil)) → c21(CONS(proper(x0), ok(nil)), PROPER(x0), PROPER(nil)) 157.90/57.17
PROPER(cons(x0, s(z0))) → c21(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 157.90/57.17
PROPER(cons(x0, cons(z0, z1))) → c21(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 157.90/57.17
PROPER(cons(x0, from(z0))) → c21(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 157.90/57.17
PROPER(cons(first(z0, z1), x1)) → c21(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(cons(0, x1)) → c21(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 157.90/57.17
PROPER(cons(nil, x1)) → c21(CONS(ok(nil), proper(x1)), PROPER(nil), PROPER(x1)) 157.90/57.17
PROPER(cons(s(z0), x1)) → c21(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 157.90/57.17
PROPER(cons(cons(z0, z1), x1)) → c21(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(cons(from(z0), x1)) → c21(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
157.90/57.17
157.90/57.17

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil) 157.90/57.17
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 157.90/57.17
active(from(z0)) → mark(cons(z0, from(s(z0)))) 157.90/57.17
active(first(z0, z1)) → first(active(z0), z1) 157.90/57.17
active(first(z0, z1)) → first(z0, active(z1)) 157.90/57.17
active(s(z0)) → s(active(z0)) 157.90/57.17
active(cons(z0, z1)) → cons(active(z0), z1) 157.90/57.17
active(from(z0)) → from(active(z0)) 157.90/57.17
first(mark(z0), z1) → mark(first(z0, z1)) 157.90/57.17
first(z0, mark(z1)) → mark(first(z0, z1)) 157.90/57.17
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 157.90/57.17
s(mark(z0)) → mark(s(z0)) 157.90/57.17
s(ok(z0)) → ok(s(z0)) 157.90/57.17
cons(mark(z0), z1) → mark(cons(z0, z1)) 157.90/57.17
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 157.90/57.17
from(mark(z0)) → mark(from(z0)) 157.90/57.17
from(ok(z0)) → ok(from(z0)) 157.90/57.17
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 157.90/57.17
proper(0) → ok(0) 157.90/57.17
proper(nil) → ok(nil) 157.90/57.17
proper(s(z0)) → s(proper(z0)) 157.90/57.17
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 157.90/57.17
proper(from(z0)) → from(proper(z0)) 157.90/57.17
top(mark(z0)) → top(proper(z0)) 157.90/57.17
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(first(z0, z1)) → c3(FIRST(active(z0), z1), ACTIVE(z0)) 157.90/57.17
ACTIVE(first(z0, z1)) → c4(FIRST(z0, active(z1)), ACTIVE(z1)) 157.90/57.17
FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.90/57.17
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.90/57.17
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.90/57.17
S(mark(z0)) → c11(S(z0)) 157.90/57.17
S(ok(z0)) → c12(S(z0)) 157.90/57.17
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.90/57.17
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.90/57.17
FROM(mark(z0)) → c15(FROM(z0)) 157.90/57.17
FROM(ok(z0)) → c16(FROM(z0)) 157.90/57.17
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 157.90/57.17
TOP(mark(z0)) → c23(TOP(proper(z0)), PROPER(z0)) 157.90/57.17
TOP(ok(z0)) → c24(TOP(active(z0)), ACTIVE(z0)) 157.90/57.17
ACTIVE(from(z0)) → c2(S(z0)) 157.90/57.17
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 157.90/57.17
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 157.90/57.17
ACTIVE(s(first(0, z0))) → c5(S(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c5(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(s(first(z0, z1))) → c5(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(s(first(z0, z1))) → c5(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(from(first(0, z0))) → c7(FROM(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c7(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(from(from(z0))) → c7(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(from(first(z0, z1))) → c7(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(from(first(z0, z1))) → c7(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(from(s(z0))) → c7(FROM(s(active(z0))), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(from(cons(z0, z1))) → c7(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(from(from(z0))) → c7(FROM(from(active(z0))), ACTIVE(from(z0))) 157.90/57.17
PROPER(first(x0, first(z0, z1))) → c17(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 157.90/57.17
PROPER(first(x0, s(z0))) → c17(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 157.90/57.17
PROPER(first(x0, cons(z0, z1))) → c17(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 157.90/57.17
PROPER(first(x0, from(z0))) → c17(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 157.90/57.17
PROPER(first(first(z0, z1), x1)) → c17(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(first(s(z0), x1)) → c17(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 157.90/57.17
PROPER(first(cons(z0, z1), x1)) → c17(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(first(from(z0), x1)) → c17(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 157.90/57.17
PROPER(first(x0, 0)) → c17(FIRST(proper(x0), ok(0)), PROPER(x0)) 157.90/57.17
PROPER(first(x0, nil)) → c17(FIRST(proper(x0), ok(nil)), PROPER(x0)) 157.90/57.17
PROPER(first(0, x1)) → c17(FIRST(ok(0), proper(x1)), PROPER(x1)) 157.90/57.17
PROPER(first(nil, x1)) → c17(FIRST(ok(nil), proper(x1)), PROPER(x1)) 157.90/57.17
PROPER(s(first(z0, z1))) → c20(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 157.90/57.17
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 157.90/57.17
PROPER(s(cons(z0, z1))) → c20(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 157.90/57.17
PROPER(s(from(z0))) → c20(S(from(proper(z0))), PROPER(from(z0))) 157.90/57.17
PROPER(s(0)) → c20(S(ok(0))) 157.90/57.17
PROPER(s(nil)) → c20(S(ok(nil))) 157.90/57.17
PROPER(cons(x0, first(z0, z1))) → c21(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 157.90/57.17
PROPER(cons(x0, 0)) → c21(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 157.90/57.17
PROPER(cons(x0, nil)) → c21(CONS(proper(x0), ok(nil)), PROPER(x0), PROPER(nil)) 157.90/57.17
PROPER(cons(x0, s(z0))) → c21(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 157.90/57.17
PROPER(cons(x0, cons(z0, z1))) → c21(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 157.90/57.17
PROPER(cons(x0, from(z0))) → c21(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 157.90/57.17
PROPER(cons(first(z0, z1), x1)) → c21(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(cons(0, x1)) → c21(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 157.90/57.17
PROPER(cons(nil, x1)) → c21(CONS(ok(nil), proper(x1)), PROPER(nil), PROPER(x1)) 157.90/57.17
PROPER(cons(s(z0), x1)) → c21(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 157.90/57.17
PROPER(cons(cons(z0, z1), x1)) → c21(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(cons(from(z0), x1)) → c21(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
S tuples:

ACTIVE(first(z0, z1)) → c3(FIRST(active(z0), z1), ACTIVE(z0)) 157.90/57.17
ACTIVE(first(z0, z1)) → c4(FIRST(z0, active(z1)), ACTIVE(z1)) 157.90/57.17
FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.90/57.17
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.90/57.17
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.90/57.17
S(mark(z0)) → c11(S(z0)) 157.90/57.17
S(ok(z0)) → c12(S(z0)) 157.90/57.17
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.90/57.17
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.90/57.17
FROM(mark(z0)) → c15(FROM(z0)) 157.90/57.17
FROM(ok(z0)) → c16(FROM(z0)) 157.90/57.17
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 157.90/57.17
TOP(ok(z0)) → c24(TOP(active(z0)), ACTIVE(z0)) 157.90/57.17
ACTIVE(from(z0)) → c2(S(z0)) 157.90/57.17
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 157.90/57.17
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 157.90/57.17
ACTIVE(s(first(0, z0))) → c5(S(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c5(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(s(first(z0, z1))) → c5(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(s(first(z0, z1))) → c5(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(from(first(0, z0))) → c7(FROM(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c7(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(from(from(z0))) → c7(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(from(first(z0, z1))) → c7(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(from(first(z0, z1))) → c7(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(from(s(z0))) → c7(FROM(s(active(z0))), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(from(cons(z0, z1))) → c7(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(from(from(z0))) → c7(FROM(from(active(z0))), ACTIVE(from(z0))) 157.90/57.17
PROPER(first(x0, first(z0, z1))) → c17(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 157.90/57.17
PROPER(first(x0, s(z0))) → c17(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 157.90/57.17
PROPER(first(x0, cons(z0, z1))) → c17(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 157.90/57.17
PROPER(first(x0, from(z0))) → c17(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 157.90/57.17
PROPER(first(first(z0, z1), x1)) → c17(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(first(s(z0), x1)) → c17(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 157.90/57.17
PROPER(first(cons(z0, z1), x1)) → c17(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(first(from(z0), x1)) → c17(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 157.90/57.17
PROPER(first(x0, 0)) → c17(FIRST(proper(x0), ok(0)), PROPER(x0)) 157.90/57.17
PROPER(first(x0, nil)) → c17(FIRST(proper(x0), ok(nil)), PROPER(x0)) 157.90/57.17
PROPER(first(0, x1)) → c17(FIRST(ok(0), proper(x1)), PROPER(x1)) 157.90/57.17
PROPER(first(nil, x1)) → c17(FIRST(ok(nil), proper(x1)), PROPER(x1)) 157.90/57.17
PROPER(s(first(z0, z1))) → c20(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 157.90/57.17
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 157.90/57.17
PROPER(s(cons(z0, z1))) → c20(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 157.90/57.17
PROPER(s(from(z0))) → c20(S(from(proper(z0))), PROPER(from(z0))) 157.90/57.17
PROPER(s(0)) → c20(S(ok(0))) 157.90/57.17
PROPER(s(nil)) → c20(S(ok(nil))) 157.90/57.17
PROPER(cons(x0, first(z0, z1))) → c21(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 157.90/57.17
PROPER(cons(x0, 0)) → c21(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 157.90/57.17
PROPER(cons(x0, nil)) → c21(CONS(proper(x0), ok(nil)), PROPER(x0), PROPER(nil)) 157.90/57.17
PROPER(cons(x0, s(z0))) → c21(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 157.90/57.17
PROPER(cons(x0, cons(z0, z1))) → c21(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 157.90/57.17
PROPER(cons(x0, from(z0))) → c21(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 157.90/57.17
PROPER(cons(first(z0, z1), x1)) → c21(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(cons(0, x1)) → c21(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 157.90/57.17
PROPER(cons(nil, x1)) → c21(CONS(ok(nil), proper(x1)), PROPER(nil), PROPER(x1)) 157.90/57.17
PROPER(cons(s(z0), x1)) → c21(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 157.90/57.17
PROPER(cons(cons(z0, z1), x1)) → c21(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(cons(from(z0), x1)) → c21(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
K tuples:

TOP(mark(z0)) → c23(TOP(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, first, s, cons, from, proper, top

Defined Pair Symbols:

ACTIVE, FIRST, S, CONS, FROM, PROPER, TOP

Compound Symbols:

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

157.90/57.17
157.90/57.17

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

Removed 4 trailing tuple parts
157.90/57.17
157.90/57.17

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil) 157.90/57.17
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 157.90/57.17
active(from(z0)) → mark(cons(z0, from(s(z0)))) 157.90/57.17
active(first(z0, z1)) → first(active(z0), z1) 157.90/57.17
active(first(z0, z1)) → first(z0, active(z1)) 157.90/57.17
active(s(z0)) → s(active(z0)) 157.90/57.17
active(cons(z0, z1)) → cons(active(z0), z1) 157.90/57.17
active(from(z0)) → from(active(z0)) 157.90/57.17
first(mark(z0), z1) → mark(first(z0, z1)) 157.90/57.17
first(z0, mark(z1)) → mark(first(z0, z1)) 157.90/57.17
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 157.90/57.17
s(mark(z0)) → mark(s(z0)) 157.90/57.17
s(ok(z0)) → ok(s(z0)) 157.90/57.17
cons(mark(z0), z1) → mark(cons(z0, z1)) 157.90/57.17
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 157.90/57.17
from(mark(z0)) → mark(from(z0)) 157.90/57.17
from(ok(z0)) → ok(from(z0)) 157.90/57.17
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 157.90/57.17
proper(0) → ok(0) 157.90/57.17
proper(nil) → ok(nil) 157.90/57.17
proper(s(z0)) → s(proper(z0)) 157.90/57.17
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 157.90/57.17
proper(from(z0)) → from(proper(z0)) 157.90/57.17
top(mark(z0)) → top(proper(z0)) 157.90/57.17
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(first(z0, z1)) → c3(FIRST(active(z0), z1), ACTIVE(z0)) 157.90/57.17
ACTIVE(first(z0, z1)) → c4(FIRST(z0, active(z1)), ACTIVE(z1)) 157.90/57.17
FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.90/57.17
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.90/57.17
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.90/57.17
S(mark(z0)) → c11(S(z0)) 157.90/57.17
S(ok(z0)) → c12(S(z0)) 157.90/57.17
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.90/57.17
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.90/57.17
FROM(mark(z0)) → c15(FROM(z0)) 157.90/57.17
FROM(ok(z0)) → c16(FROM(z0)) 157.90/57.17
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 157.90/57.17
TOP(mark(z0)) → c23(TOP(proper(z0)), PROPER(z0)) 157.90/57.17
TOP(ok(z0)) → c24(TOP(active(z0)), ACTIVE(z0)) 157.90/57.17
ACTIVE(from(z0)) → c2(S(z0)) 157.90/57.17
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 157.90/57.17
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 157.90/57.17
ACTIVE(s(first(0, z0))) → c5(S(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c5(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(s(first(z0, z1))) → c5(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(s(first(z0, z1))) → c5(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(from(first(0, z0))) → c7(FROM(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c7(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(from(from(z0))) → c7(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(from(first(z0, z1))) → c7(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(from(first(z0, z1))) → c7(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(from(s(z0))) → c7(FROM(s(active(z0))), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(from(cons(z0, z1))) → c7(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(from(from(z0))) → c7(FROM(from(active(z0))), ACTIVE(from(z0))) 157.90/57.17
PROPER(first(x0, first(z0, z1))) → c17(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 157.90/57.17
PROPER(first(x0, s(z0))) → c17(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 157.90/57.17
PROPER(first(x0, cons(z0, z1))) → c17(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 157.90/57.17
PROPER(first(x0, from(z0))) → c17(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 157.90/57.17
PROPER(first(first(z0, z1), x1)) → c17(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(first(s(z0), x1)) → c17(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 157.90/57.17
PROPER(first(cons(z0, z1), x1)) → c17(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(first(from(z0), x1)) → c17(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 157.90/57.17
PROPER(first(x0, 0)) → c17(FIRST(proper(x0), ok(0)), PROPER(x0)) 157.90/57.17
PROPER(first(x0, nil)) → c17(FIRST(proper(x0), ok(nil)), PROPER(x0)) 157.90/57.17
PROPER(first(0, x1)) → c17(FIRST(ok(0), proper(x1)), PROPER(x1)) 157.90/57.17
PROPER(first(nil, x1)) → c17(FIRST(ok(nil), proper(x1)), PROPER(x1)) 157.90/57.17
PROPER(s(first(z0, z1))) → c20(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 157.90/57.17
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 157.90/57.17
PROPER(s(cons(z0, z1))) → c20(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 157.90/57.17
PROPER(s(from(z0))) → c20(S(from(proper(z0))), PROPER(from(z0))) 157.90/57.17
PROPER(s(0)) → c20(S(ok(0))) 157.90/57.17
PROPER(s(nil)) → c20(S(ok(nil))) 157.90/57.17
PROPER(cons(x0, first(z0, z1))) → c21(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 157.90/57.17
PROPER(cons(x0, s(z0))) → c21(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 157.90/57.17
PROPER(cons(x0, cons(z0, z1))) → c21(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 157.90/57.17
PROPER(cons(x0, from(z0))) → c21(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 157.90/57.17
PROPER(cons(first(z0, z1), x1)) → c21(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(cons(s(z0), x1)) → c21(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 157.90/57.17
PROPER(cons(cons(z0, z1), x1)) → c21(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(cons(from(z0), x1)) → c21(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 157.90/57.17
PROPER(cons(x0, 0)) → c21(CONS(proper(x0), ok(0)), PROPER(x0)) 157.90/57.17
PROPER(cons(x0, nil)) → c21(CONS(proper(x0), ok(nil)), PROPER(x0)) 157.90/57.17
PROPER(cons(0, x1)) → c21(CONS(ok(0), proper(x1)), PROPER(x1)) 157.90/57.17
PROPER(cons(nil, x1)) → c21(CONS(ok(nil), proper(x1)), PROPER(x1))
S tuples:

ACTIVE(first(z0, z1)) → c3(FIRST(active(z0), z1), ACTIVE(z0)) 157.90/57.17
ACTIVE(first(z0, z1)) → c4(FIRST(z0, active(z1)), ACTIVE(z1)) 157.90/57.17
FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.90/57.17
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.90/57.17
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.90/57.17
S(mark(z0)) → c11(S(z0)) 157.90/57.17
S(ok(z0)) → c12(S(z0)) 157.90/57.17
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.90/57.17
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.90/57.17
FROM(mark(z0)) → c15(FROM(z0)) 157.90/57.17
FROM(ok(z0)) → c16(FROM(z0)) 157.90/57.17
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 157.90/57.17
TOP(ok(z0)) → c24(TOP(active(z0)), ACTIVE(z0)) 157.90/57.17
ACTIVE(from(z0)) → c2(S(z0)) 157.90/57.17
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 157.90/57.17
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 157.90/57.17
ACTIVE(s(first(0, z0))) → c5(S(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c5(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(s(first(z0, z1))) → c5(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(s(first(z0, z1))) → c5(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(from(first(0, z0))) → c7(FROM(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c7(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(from(from(z0))) → c7(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(from(first(z0, z1))) → c7(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(from(first(z0, z1))) → c7(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(from(s(z0))) → c7(FROM(s(active(z0))), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(from(cons(z0, z1))) → c7(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(from(from(z0))) → c7(FROM(from(active(z0))), ACTIVE(from(z0))) 157.90/57.17
PROPER(first(x0, first(z0, z1))) → c17(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 157.90/57.17
PROPER(first(x0, s(z0))) → c17(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 157.90/57.17
PROPER(first(x0, cons(z0, z1))) → c17(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 157.90/57.17
PROPER(first(x0, from(z0))) → c17(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 157.90/57.17
PROPER(first(first(z0, z1), x1)) → c17(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(first(s(z0), x1)) → c17(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 157.90/57.17
PROPER(first(cons(z0, z1), x1)) → c17(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(first(from(z0), x1)) → c17(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 157.90/57.17
PROPER(first(x0, 0)) → c17(FIRST(proper(x0), ok(0)), PROPER(x0)) 157.90/57.17
PROPER(first(x0, nil)) → c17(FIRST(proper(x0), ok(nil)), PROPER(x0)) 157.90/57.17
PROPER(first(0, x1)) → c17(FIRST(ok(0), proper(x1)), PROPER(x1)) 157.90/57.17
PROPER(first(nil, x1)) → c17(FIRST(ok(nil), proper(x1)), PROPER(x1)) 157.90/57.17
PROPER(s(first(z0, z1))) → c20(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 157.90/57.17
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 157.90/57.17
PROPER(s(cons(z0, z1))) → c20(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 157.90/57.17
PROPER(s(from(z0))) → c20(S(from(proper(z0))), PROPER(from(z0))) 157.90/57.17
PROPER(s(0)) → c20(S(ok(0))) 157.90/57.17
PROPER(s(nil)) → c20(S(ok(nil))) 157.90/57.17
PROPER(cons(x0, first(z0, z1))) → c21(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 157.90/57.17
PROPER(cons(x0, s(z0))) → c21(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 157.90/57.17
PROPER(cons(x0, cons(z0, z1))) → c21(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 157.90/57.17
PROPER(cons(x0, from(z0))) → c21(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 157.90/57.17
PROPER(cons(first(z0, z1), x1)) → c21(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(cons(s(z0), x1)) → c21(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 157.90/57.17
PROPER(cons(cons(z0, z1), x1)) → c21(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(cons(from(z0), x1)) → c21(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 157.90/57.17
PROPER(cons(x0, 0)) → c21(CONS(proper(x0), ok(0)), PROPER(x0)) 157.90/57.17
PROPER(cons(x0, nil)) → c21(CONS(proper(x0), ok(nil)), PROPER(x0)) 157.90/57.17
PROPER(cons(0, x1)) → c21(CONS(ok(0), proper(x1)), PROPER(x1)) 157.90/57.17
PROPER(cons(nil, x1)) → c21(CONS(ok(nil), proper(x1)), PROPER(x1))
K tuples:

TOP(mark(z0)) → c23(TOP(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, first, s, cons, from, proper, top

Defined Pair Symbols:

ACTIVE, FIRST, S, CONS, FROM, PROPER, TOP

Compound Symbols:

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

157.90/57.17
157.90/57.17

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

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

PROPER(from(first(z0, z1))) → c22(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 157.90/57.17
PROPER(from(0)) → c22(FROM(ok(0)), PROPER(0)) 157.90/57.17
PROPER(from(nil)) → c22(FROM(ok(nil)), PROPER(nil)) 157.90/57.17
PROPER(from(s(z0))) → c22(FROM(s(proper(z0))), PROPER(s(z0))) 157.90/57.17
PROPER(from(cons(z0, z1))) → c22(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 157.90/57.17
PROPER(from(from(z0))) → c22(FROM(from(proper(z0))), PROPER(from(z0)))
157.90/57.17
157.90/57.17

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil) 157.90/57.17
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 157.90/57.17
active(from(z0)) → mark(cons(z0, from(s(z0)))) 157.90/57.17
active(first(z0, z1)) → first(active(z0), z1) 157.90/57.17
active(first(z0, z1)) → first(z0, active(z1)) 157.90/57.17
active(s(z0)) → s(active(z0)) 157.90/57.17
active(cons(z0, z1)) → cons(active(z0), z1) 157.90/57.17
active(from(z0)) → from(active(z0)) 157.90/57.17
first(mark(z0), z1) → mark(first(z0, z1)) 157.90/57.17
first(z0, mark(z1)) → mark(first(z0, z1)) 157.90/57.17
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 157.90/57.17
s(mark(z0)) → mark(s(z0)) 157.90/57.17
s(ok(z0)) → ok(s(z0)) 157.90/57.17
cons(mark(z0), z1) → mark(cons(z0, z1)) 157.90/57.17
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 157.90/57.17
from(mark(z0)) → mark(from(z0)) 157.90/57.17
from(ok(z0)) → ok(from(z0)) 157.90/57.17
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 157.90/57.17
proper(0) → ok(0) 157.90/57.17
proper(nil) → ok(nil) 157.90/57.17
proper(s(z0)) → s(proper(z0)) 157.90/57.17
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 157.90/57.17
proper(from(z0)) → from(proper(z0)) 157.90/57.17
top(mark(z0)) → top(proper(z0)) 157.90/57.17
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(first(z0, z1)) → c3(FIRST(active(z0), z1), ACTIVE(z0)) 157.90/57.17
ACTIVE(first(z0, z1)) → c4(FIRST(z0, active(z1)), ACTIVE(z1)) 157.90/57.17
FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.90/57.17
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.90/57.17
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.90/57.17
S(mark(z0)) → c11(S(z0)) 157.90/57.17
S(ok(z0)) → c12(S(z0)) 157.90/57.17
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.90/57.17
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.90/57.17
FROM(mark(z0)) → c15(FROM(z0)) 157.90/57.17
FROM(ok(z0)) → c16(FROM(z0)) 157.90/57.17
TOP(mark(z0)) → c23(TOP(proper(z0)), PROPER(z0)) 157.90/57.17
TOP(ok(z0)) → c24(TOP(active(z0)), ACTIVE(z0)) 157.90/57.17
ACTIVE(from(z0)) → c2(S(z0)) 157.90/57.17
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 157.90/57.17
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 157.90/57.17
ACTIVE(s(first(0, z0))) → c5(S(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c5(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(s(first(z0, z1))) → c5(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(s(first(z0, z1))) → c5(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(from(first(0, z0))) → c7(FROM(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c7(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(from(from(z0))) → c7(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(from(first(z0, z1))) → c7(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(from(first(z0, z1))) → c7(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(from(s(z0))) → c7(FROM(s(active(z0))), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(from(cons(z0, z1))) → c7(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(from(from(z0))) → c7(FROM(from(active(z0))), ACTIVE(from(z0))) 157.90/57.17
PROPER(first(x0, first(z0, z1))) → c17(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 157.90/57.17
PROPER(first(x0, s(z0))) → c17(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 157.90/57.17
PROPER(first(x0, cons(z0, z1))) → c17(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 157.90/57.17
PROPER(first(x0, from(z0))) → c17(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 157.90/57.17
PROPER(first(first(z0, z1), x1)) → c17(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(first(s(z0), x1)) → c17(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 157.90/57.17
PROPER(first(cons(z0, z1), x1)) → c17(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(first(from(z0), x1)) → c17(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 157.90/57.17
PROPER(first(x0, 0)) → c17(FIRST(proper(x0), ok(0)), PROPER(x0)) 157.90/57.17
PROPER(first(x0, nil)) → c17(FIRST(proper(x0), ok(nil)), PROPER(x0)) 157.90/57.17
PROPER(first(0, x1)) → c17(FIRST(ok(0), proper(x1)), PROPER(x1)) 157.90/57.17
PROPER(first(nil, x1)) → c17(FIRST(ok(nil), proper(x1)), PROPER(x1)) 157.90/57.17
PROPER(s(first(z0, z1))) → c20(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 157.90/57.17
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 157.90/57.17
PROPER(s(cons(z0, z1))) → c20(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 157.90/57.17
PROPER(s(from(z0))) → c20(S(from(proper(z0))), PROPER(from(z0))) 157.90/57.17
PROPER(s(0)) → c20(S(ok(0))) 157.90/57.17
PROPER(s(nil)) → c20(S(ok(nil))) 157.90/57.17
PROPER(cons(x0, first(z0, z1))) → c21(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 157.90/57.17
PROPER(cons(x0, s(z0))) → c21(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 157.90/57.17
PROPER(cons(x0, cons(z0, z1))) → c21(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 157.90/57.17
PROPER(cons(x0, from(z0))) → c21(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 157.90/57.17
PROPER(cons(first(z0, z1), x1)) → c21(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(cons(s(z0), x1)) → c21(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 157.90/57.17
PROPER(cons(cons(z0, z1), x1)) → c21(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(cons(from(z0), x1)) → c21(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 157.90/57.17
PROPER(cons(x0, 0)) → c21(CONS(proper(x0), ok(0)), PROPER(x0)) 157.90/57.17
PROPER(cons(x0, nil)) → c21(CONS(proper(x0), ok(nil)), PROPER(x0)) 157.90/57.17
PROPER(cons(0, x1)) → c21(CONS(ok(0), proper(x1)), PROPER(x1)) 157.90/57.17
PROPER(cons(nil, x1)) → c21(CONS(ok(nil), proper(x1)), PROPER(x1)) 157.90/57.17
PROPER(from(first(z0, z1))) → c22(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 157.90/57.17
PROPER(from(0)) → c22(FROM(ok(0)), PROPER(0)) 157.90/57.17
PROPER(from(nil)) → c22(FROM(ok(nil)), PROPER(nil)) 157.90/57.17
PROPER(from(s(z0))) → c22(FROM(s(proper(z0))), PROPER(s(z0))) 157.90/57.17
PROPER(from(cons(z0, z1))) → c22(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 157.90/57.17
PROPER(from(from(z0))) → c22(FROM(from(proper(z0))), PROPER(from(z0)))
S tuples:

ACTIVE(first(z0, z1)) → c3(FIRST(active(z0), z1), ACTIVE(z0)) 157.90/57.17
ACTIVE(first(z0, z1)) → c4(FIRST(z0, active(z1)), ACTIVE(z1)) 157.90/57.17
FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.90/57.17
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.90/57.17
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.90/57.17
S(mark(z0)) → c11(S(z0)) 157.90/57.17
S(ok(z0)) → c12(S(z0)) 157.90/57.17
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.90/57.17
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.90/57.17
FROM(mark(z0)) → c15(FROM(z0)) 157.90/57.17
FROM(ok(z0)) → c16(FROM(z0)) 157.90/57.17
TOP(ok(z0)) → c24(TOP(active(z0)), ACTIVE(z0)) 157.90/57.17
ACTIVE(from(z0)) → c2(S(z0)) 157.90/57.17
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 157.90/57.17
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 157.90/57.17
ACTIVE(s(first(0, z0))) → c5(S(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c5(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(s(first(z0, z1))) → c5(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(s(first(z0, z1))) → c5(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(from(first(0, z0))) → c7(FROM(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c7(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(from(from(z0))) → c7(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(from(first(z0, z1))) → c7(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(from(first(z0, z1))) → c7(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(from(s(z0))) → c7(FROM(s(active(z0))), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(from(cons(z0, z1))) → c7(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(from(from(z0))) → c7(FROM(from(active(z0))), ACTIVE(from(z0))) 157.90/57.17
PROPER(first(x0, first(z0, z1))) → c17(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 157.90/57.17
PROPER(first(x0, s(z0))) → c17(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 157.90/57.17
PROPER(first(x0, cons(z0, z1))) → c17(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 157.90/57.17
PROPER(first(x0, from(z0))) → c17(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 157.90/57.17
PROPER(first(first(z0, z1), x1)) → c17(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(first(s(z0), x1)) → c17(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 157.90/57.17
PROPER(first(cons(z0, z1), x1)) → c17(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(first(from(z0), x1)) → c17(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 157.90/57.17
PROPER(first(x0, 0)) → c17(FIRST(proper(x0), ok(0)), PROPER(x0)) 157.90/57.17
PROPER(first(x0, nil)) → c17(FIRST(proper(x0), ok(nil)), PROPER(x0)) 157.90/57.17
PROPER(first(0, x1)) → c17(FIRST(ok(0), proper(x1)), PROPER(x1)) 157.90/57.17
PROPER(first(nil, x1)) → c17(FIRST(ok(nil), proper(x1)), PROPER(x1)) 157.90/57.17
PROPER(s(first(z0, z1))) → c20(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 157.90/57.17
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 157.90/57.17
PROPER(s(cons(z0, z1))) → c20(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 157.90/57.17
PROPER(s(from(z0))) → c20(S(from(proper(z0))), PROPER(from(z0))) 157.90/57.17
PROPER(s(0)) → c20(S(ok(0))) 157.90/57.17
PROPER(s(nil)) → c20(S(ok(nil))) 157.90/57.17
PROPER(cons(x0, first(z0, z1))) → c21(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 157.90/57.17
PROPER(cons(x0, s(z0))) → c21(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 157.90/57.17
PROPER(cons(x0, cons(z0, z1))) → c21(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 157.90/57.17
PROPER(cons(x0, from(z0))) → c21(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 157.90/57.17
PROPER(cons(first(z0, z1), x1)) → c21(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(cons(s(z0), x1)) → c21(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 157.90/57.17
PROPER(cons(cons(z0, z1), x1)) → c21(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(cons(from(z0), x1)) → c21(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 157.90/57.17
PROPER(cons(x0, 0)) → c21(CONS(proper(x0), ok(0)), PROPER(x0)) 157.90/57.17
PROPER(cons(x0, nil)) → c21(CONS(proper(x0), ok(nil)), PROPER(x0)) 157.90/57.17
PROPER(cons(0, x1)) → c21(CONS(ok(0), proper(x1)), PROPER(x1)) 157.90/57.17
PROPER(cons(nil, x1)) → c21(CONS(ok(nil), proper(x1)), PROPER(x1)) 157.90/57.17
PROPER(from(first(z0, z1))) → c22(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 157.90/57.17
PROPER(from(0)) → c22(FROM(ok(0)), PROPER(0)) 157.90/57.17
PROPER(from(nil)) → c22(FROM(ok(nil)), PROPER(nil)) 157.90/57.17
PROPER(from(s(z0))) → c22(FROM(s(proper(z0))), PROPER(s(z0))) 157.90/57.17
PROPER(from(cons(z0, z1))) → c22(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 157.90/57.17
PROPER(from(from(z0))) → c22(FROM(from(proper(z0))), PROPER(from(z0)))
K tuples:

TOP(mark(z0)) → c23(TOP(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, first, s, cons, from, proper, top

Defined Pair Symbols:

ACTIVE, FIRST, S, CONS, FROM, TOP, PROPER

Compound Symbols:

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

157.90/57.17
157.90/57.17

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

Removed 2 trailing tuple parts
157.90/57.17
157.90/57.17

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil) 157.90/57.17
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 157.90/57.17
active(from(z0)) → mark(cons(z0, from(s(z0)))) 157.90/57.17
active(first(z0, z1)) → first(active(z0), z1) 157.90/57.17
active(first(z0, z1)) → first(z0, active(z1)) 157.90/57.17
active(s(z0)) → s(active(z0)) 157.90/57.17
active(cons(z0, z1)) → cons(active(z0), z1) 157.90/57.17
active(from(z0)) → from(active(z0)) 157.90/57.17
first(mark(z0), z1) → mark(first(z0, z1)) 157.90/57.17
first(z0, mark(z1)) → mark(first(z0, z1)) 157.90/57.17
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 157.90/57.17
s(mark(z0)) → mark(s(z0)) 157.90/57.17
s(ok(z0)) → ok(s(z0)) 157.90/57.17
cons(mark(z0), z1) → mark(cons(z0, z1)) 157.90/57.17
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 157.90/57.17
from(mark(z0)) → mark(from(z0)) 157.90/57.17
from(ok(z0)) → ok(from(z0)) 157.90/57.17
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 157.90/57.17
proper(0) → ok(0) 157.90/57.17
proper(nil) → ok(nil) 157.90/57.17
proper(s(z0)) → s(proper(z0)) 157.90/57.17
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 157.90/57.17
proper(from(z0)) → from(proper(z0)) 157.90/57.17
top(mark(z0)) → top(proper(z0)) 157.90/57.17
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(first(z0, z1)) → c3(FIRST(active(z0), z1), ACTIVE(z0)) 157.90/57.17
ACTIVE(first(z0, z1)) → c4(FIRST(z0, active(z1)), ACTIVE(z1)) 157.90/57.17
FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.90/57.17
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.90/57.17
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.90/57.17
S(mark(z0)) → c11(S(z0)) 157.90/57.17
S(ok(z0)) → c12(S(z0)) 157.90/57.17
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.90/57.17
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.90/57.17
FROM(mark(z0)) → c15(FROM(z0)) 157.90/57.17
FROM(ok(z0)) → c16(FROM(z0)) 157.90/57.17
TOP(mark(z0)) → c23(TOP(proper(z0)), PROPER(z0)) 157.90/57.17
TOP(ok(z0)) → c24(TOP(active(z0)), ACTIVE(z0)) 157.90/57.17
ACTIVE(from(z0)) → c2(S(z0)) 157.90/57.17
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 157.90/57.17
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 157.90/57.17
ACTIVE(s(first(0, z0))) → c5(S(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c5(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(s(first(z0, z1))) → c5(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(s(first(z0, z1))) → c5(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(from(first(0, z0))) → c7(FROM(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c7(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(from(from(z0))) → c7(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(from(first(z0, z1))) → c7(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(from(first(z0, z1))) → c7(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(from(s(z0))) → c7(FROM(s(active(z0))), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(from(cons(z0, z1))) → c7(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(from(from(z0))) → c7(FROM(from(active(z0))), ACTIVE(from(z0))) 157.90/57.17
PROPER(first(x0, first(z0, z1))) → c17(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 157.90/57.17
PROPER(first(x0, s(z0))) → c17(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 157.90/57.17
PROPER(first(x0, cons(z0, z1))) → c17(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 157.90/57.17
PROPER(first(x0, from(z0))) → c17(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 157.90/57.17
PROPER(first(first(z0, z1), x1)) → c17(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(first(s(z0), x1)) → c17(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 157.90/57.17
PROPER(first(cons(z0, z1), x1)) → c17(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(first(from(z0), x1)) → c17(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 157.90/57.17
PROPER(first(x0, 0)) → c17(FIRST(proper(x0), ok(0)), PROPER(x0)) 157.90/57.17
PROPER(first(x0, nil)) → c17(FIRST(proper(x0), ok(nil)), PROPER(x0)) 157.90/57.17
PROPER(first(0, x1)) → c17(FIRST(ok(0), proper(x1)), PROPER(x1)) 157.90/57.17
PROPER(first(nil, x1)) → c17(FIRST(ok(nil), proper(x1)), PROPER(x1)) 157.90/57.17
PROPER(s(first(z0, z1))) → c20(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 157.90/57.17
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 157.90/57.17
PROPER(s(cons(z0, z1))) → c20(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 157.90/57.17
PROPER(s(from(z0))) → c20(S(from(proper(z0))), PROPER(from(z0))) 157.90/57.17
PROPER(s(0)) → c20(S(ok(0))) 157.90/57.17
PROPER(s(nil)) → c20(S(ok(nil))) 157.90/57.17
PROPER(cons(x0, first(z0, z1))) → c21(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 157.90/57.17
PROPER(cons(x0, s(z0))) → c21(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 157.90/57.17
PROPER(cons(x0, cons(z0, z1))) → c21(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 157.90/57.17
PROPER(cons(x0, from(z0))) → c21(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 157.90/57.17
PROPER(cons(first(z0, z1), x1)) → c21(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(cons(s(z0), x1)) → c21(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 157.90/57.17
PROPER(cons(cons(z0, z1), x1)) → c21(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(cons(from(z0), x1)) → c21(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 157.90/57.17
PROPER(cons(x0, 0)) → c21(CONS(proper(x0), ok(0)), PROPER(x0)) 157.90/57.17
PROPER(cons(x0, nil)) → c21(CONS(proper(x0), ok(nil)), PROPER(x0)) 157.90/57.17
PROPER(cons(0, x1)) → c21(CONS(ok(0), proper(x1)), PROPER(x1)) 157.90/57.17
PROPER(cons(nil, x1)) → c21(CONS(ok(nil), proper(x1)), PROPER(x1)) 157.90/57.17
PROPER(from(first(z0, z1))) → c22(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 157.90/57.17
PROPER(from(s(z0))) → c22(FROM(s(proper(z0))), PROPER(s(z0))) 157.90/57.17
PROPER(from(cons(z0, z1))) → c22(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 157.90/57.17
PROPER(from(from(z0))) → c22(FROM(from(proper(z0))), PROPER(from(z0))) 157.90/57.17
PROPER(from(0)) → c22(FROM(ok(0))) 157.90/57.17
PROPER(from(nil)) → c22(FROM(ok(nil)))
S tuples:

ACTIVE(first(z0, z1)) → c3(FIRST(active(z0), z1), ACTIVE(z0)) 157.90/57.17
ACTIVE(first(z0, z1)) → c4(FIRST(z0, active(z1)), ACTIVE(z1)) 157.90/57.17
FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.90/57.17
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.90/57.17
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.90/57.17
S(mark(z0)) → c11(S(z0)) 157.90/57.17
S(ok(z0)) → c12(S(z0)) 157.90/57.17
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.90/57.17
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.90/57.17
FROM(mark(z0)) → c15(FROM(z0)) 157.90/57.17
FROM(ok(z0)) → c16(FROM(z0)) 157.90/57.17
TOP(ok(z0)) → c24(TOP(active(z0)), ACTIVE(z0)) 157.90/57.17
ACTIVE(from(z0)) → c2(S(z0)) 157.90/57.17
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 157.90/57.17
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 157.90/57.17
ACTIVE(s(first(0, z0))) → c5(S(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c5(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(s(first(z0, z1))) → c5(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(s(first(z0, z1))) → c5(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(from(first(0, z0))) → c7(FROM(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.17
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c7(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.17
ACTIVE(from(from(z0))) → c7(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.17
ACTIVE(from(first(z0, z1))) → c7(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(from(first(z0, z1))) → c7(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.17
ACTIVE(from(s(z0))) → c7(FROM(s(active(z0))), ACTIVE(s(z0))) 157.90/57.17
ACTIVE(from(cons(z0, z1))) → c7(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.17
ACTIVE(from(from(z0))) → c7(FROM(from(active(z0))), ACTIVE(from(z0))) 157.90/57.17
PROPER(first(x0, first(z0, z1))) → c17(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 157.90/57.17
PROPER(first(x0, s(z0))) → c17(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 157.90/57.17
PROPER(first(x0, cons(z0, z1))) → c17(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 157.90/57.17
PROPER(first(x0, from(z0))) → c17(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 157.90/57.17
PROPER(first(first(z0, z1), x1)) → c17(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(first(s(z0), x1)) → c17(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 157.90/57.17
PROPER(first(cons(z0, z1), x1)) → c17(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(first(from(z0), x1)) → c17(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 157.90/57.17
PROPER(first(x0, 0)) → c17(FIRST(proper(x0), ok(0)), PROPER(x0)) 157.90/57.17
PROPER(first(x0, nil)) → c17(FIRST(proper(x0), ok(nil)), PROPER(x0)) 157.90/57.17
PROPER(first(0, x1)) → c17(FIRST(ok(0), proper(x1)), PROPER(x1)) 157.90/57.17
PROPER(first(nil, x1)) → c17(FIRST(ok(nil), proper(x1)), PROPER(x1)) 157.90/57.17
PROPER(s(first(z0, z1))) → c20(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 157.90/57.17
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 157.90/57.17
PROPER(s(cons(z0, z1))) → c20(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 157.90/57.17
PROPER(s(from(z0))) → c20(S(from(proper(z0))), PROPER(from(z0))) 157.90/57.17
PROPER(s(0)) → c20(S(ok(0))) 157.90/57.17
PROPER(s(nil)) → c20(S(ok(nil))) 157.90/57.17
PROPER(cons(x0, first(z0, z1))) → c21(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 157.90/57.17
PROPER(cons(x0, s(z0))) → c21(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 157.90/57.17
PROPER(cons(x0, cons(z0, z1))) → c21(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 157.90/57.17
PROPER(cons(x0, from(z0))) → c21(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 157.90/57.17
PROPER(cons(first(z0, z1), x1)) → c21(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(cons(s(z0), x1)) → c21(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 157.90/57.17
PROPER(cons(cons(z0, z1), x1)) → c21(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 157.90/57.17
PROPER(cons(from(z0), x1)) → c21(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 157.90/57.17
PROPER(cons(x0, 0)) → c21(CONS(proper(x0), ok(0)), PROPER(x0)) 157.90/57.17
PROPER(cons(x0, nil)) → c21(CONS(proper(x0), ok(nil)), PROPER(x0)) 157.90/57.18
PROPER(cons(0, x1)) → c21(CONS(ok(0), proper(x1)), PROPER(x1)) 157.90/57.18
PROPER(cons(nil, x1)) → c21(CONS(ok(nil), proper(x1)), PROPER(x1)) 157.90/57.18
PROPER(from(first(z0, z1))) → c22(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 157.90/57.18
PROPER(from(s(z0))) → c22(FROM(s(proper(z0))), PROPER(s(z0))) 157.90/57.18
PROPER(from(cons(z0, z1))) → c22(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 157.90/57.18
PROPER(from(from(z0))) → c22(FROM(from(proper(z0))), PROPER(from(z0))) 157.90/57.18
PROPER(from(0)) → c22(FROM(ok(0))) 157.90/57.18
PROPER(from(nil)) → c22(FROM(ok(nil)))
K tuples:

TOP(mark(z0)) → c23(TOP(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, first, s, cons, from, proper, top

Defined Pair Symbols:

ACTIVE, FIRST, S, CONS, FROM, TOP, PROPER

Compound Symbols:

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

157.90/57.18
157.90/57.18

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

Use narrowing to replace TOP(mark(z0)) → c23(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(first(z0, z1))) → c23(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 157.90/57.18
TOP(mark(0)) → c23(TOP(ok(0)), PROPER(0)) 157.90/57.18
TOP(mark(nil)) → c23(TOP(ok(nil)), PROPER(nil)) 157.90/57.18
TOP(mark(s(z0))) → c23(TOP(s(proper(z0))), PROPER(s(z0))) 157.90/57.18
TOP(mark(cons(z0, z1))) → c23(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 157.90/57.18
TOP(mark(from(z0))) → c23(TOP(from(proper(z0))), PROPER(from(z0)))
157.90/57.18
157.90/57.18

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil) 157.90/57.18
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 157.90/57.18
active(from(z0)) → mark(cons(z0, from(s(z0)))) 157.90/57.18
active(first(z0, z1)) → first(active(z0), z1) 157.90/57.18
active(first(z0, z1)) → first(z0, active(z1)) 157.90/57.18
active(s(z0)) → s(active(z0)) 157.90/57.18
active(cons(z0, z1)) → cons(active(z0), z1) 157.90/57.18
active(from(z0)) → from(active(z0)) 157.90/57.18
first(mark(z0), z1) → mark(first(z0, z1)) 157.90/57.18
first(z0, mark(z1)) → mark(first(z0, z1)) 157.90/57.18
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 157.90/57.18
s(mark(z0)) → mark(s(z0)) 157.90/57.18
s(ok(z0)) → ok(s(z0)) 157.90/57.18
cons(mark(z0), z1) → mark(cons(z0, z1)) 157.90/57.18
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 157.90/57.18
from(mark(z0)) → mark(from(z0)) 157.90/57.18
from(ok(z0)) → ok(from(z0)) 157.90/57.18
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 157.90/57.18
proper(0) → ok(0) 157.90/57.18
proper(nil) → ok(nil) 157.90/57.18
proper(s(z0)) → s(proper(z0)) 157.90/57.18
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 157.90/57.18
proper(from(z0)) → from(proper(z0)) 157.90/57.18
top(mark(z0)) → top(proper(z0)) 157.90/57.18
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(first(z0, z1)) → c3(FIRST(active(z0), z1), ACTIVE(z0)) 157.90/57.18
ACTIVE(first(z0, z1)) → c4(FIRST(z0, active(z1)), ACTIVE(z1)) 157.90/57.18
FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.90/57.18
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.90/57.18
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.90/57.18
S(mark(z0)) → c11(S(z0)) 157.90/57.18
S(ok(z0)) → c12(S(z0)) 157.90/57.18
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.90/57.18
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.90/57.18
FROM(mark(z0)) → c15(FROM(z0)) 157.90/57.18
FROM(ok(z0)) → c16(FROM(z0)) 157.90/57.18
TOP(ok(z0)) → c24(TOP(active(z0)), ACTIVE(z0)) 157.90/57.18
ACTIVE(from(z0)) → c2(S(z0)) 157.90/57.18
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 157.90/57.18
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 157.90/57.18
ACTIVE(s(first(0, z0))) → c5(S(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.18
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c5(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.18
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.18
ACTIVE(s(first(z0, z1))) → c5(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.18
ACTIVE(s(first(z0, z1))) → c5(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.18
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 157.90/57.18
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.18
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 157.90/57.18
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 157.90/57.18
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.18
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 157.90/57.18
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 157.90/57.18
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 157.90/57.18
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 157.90/57.18
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 157.90/57.18
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 157.90/57.18
ACTIVE(from(first(0, z0))) → c7(FROM(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.18
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c7(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.18
ACTIVE(from(from(z0))) → c7(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.18
ACTIVE(from(first(z0, z1))) → c7(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.18
ACTIVE(from(first(z0, z1))) → c7(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.18
ACTIVE(from(s(z0))) → c7(FROM(s(active(z0))), ACTIVE(s(z0))) 157.90/57.18
ACTIVE(from(cons(z0, z1))) → c7(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.18
ACTIVE(from(from(z0))) → c7(FROM(from(active(z0))), ACTIVE(from(z0))) 157.90/57.18
PROPER(first(x0, first(z0, z1))) → c17(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 157.90/57.18
PROPER(first(x0, s(z0))) → c17(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 157.90/57.18
PROPER(first(x0, cons(z0, z1))) → c17(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 157.90/57.18
PROPER(first(x0, from(z0))) → c17(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 157.90/57.18
PROPER(first(first(z0, z1), x1)) → c17(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 157.90/57.18
PROPER(first(s(z0), x1)) → c17(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 157.90/57.18
PROPER(first(cons(z0, z1), x1)) → c17(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 157.90/57.18
PROPER(first(from(z0), x1)) → c17(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 157.90/57.18
PROPER(first(x0, 0)) → c17(FIRST(proper(x0), ok(0)), PROPER(x0)) 157.90/57.18
PROPER(first(x0, nil)) → c17(FIRST(proper(x0), ok(nil)), PROPER(x0)) 157.90/57.18
PROPER(first(0, x1)) → c17(FIRST(ok(0), proper(x1)), PROPER(x1)) 157.90/57.18
PROPER(first(nil, x1)) → c17(FIRST(ok(nil), proper(x1)), PROPER(x1)) 157.90/57.18
PROPER(s(first(z0, z1))) → c20(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 157.90/57.18
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 157.90/57.18
PROPER(s(cons(z0, z1))) → c20(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 157.90/57.18
PROPER(s(from(z0))) → c20(S(from(proper(z0))), PROPER(from(z0))) 157.90/57.18
PROPER(s(0)) → c20(S(ok(0))) 157.90/57.18
PROPER(s(nil)) → c20(S(ok(nil))) 157.90/57.18
PROPER(cons(x0, first(z0, z1))) → c21(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 157.90/57.18
PROPER(cons(x0, s(z0))) → c21(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 157.90/57.18
PROPER(cons(x0, cons(z0, z1))) → c21(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 157.90/57.18
PROPER(cons(x0, from(z0))) → c21(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 157.90/57.18
PROPER(cons(first(z0, z1), x1)) → c21(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 157.90/57.18
PROPER(cons(s(z0), x1)) → c21(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 157.90/57.18
PROPER(cons(cons(z0, z1), x1)) → c21(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 157.90/57.18
PROPER(cons(from(z0), x1)) → c21(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 157.90/57.18
PROPER(cons(x0, 0)) → c21(CONS(proper(x0), ok(0)), PROPER(x0)) 157.90/57.18
PROPER(cons(x0, nil)) → c21(CONS(proper(x0), ok(nil)), PROPER(x0)) 157.90/57.18
PROPER(cons(0, x1)) → c21(CONS(ok(0), proper(x1)), PROPER(x1)) 157.90/57.18
PROPER(cons(nil, x1)) → c21(CONS(ok(nil), proper(x1)), PROPER(x1)) 157.90/57.18
PROPER(from(first(z0, z1))) → c22(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 157.90/57.18
PROPER(from(s(z0))) → c22(FROM(s(proper(z0))), PROPER(s(z0))) 157.90/57.18
PROPER(from(cons(z0, z1))) → c22(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 157.90/57.18
PROPER(from(from(z0))) → c22(FROM(from(proper(z0))), PROPER(from(z0))) 157.90/57.18
PROPER(from(0)) → c22(FROM(ok(0))) 157.90/57.18
PROPER(from(nil)) → c22(FROM(ok(nil))) 157.90/57.18
TOP(mark(first(z0, z1))) → c23(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 157.90/57.18
TOP(mark(0)) → c23(TOP(ok(0)), PROPER(0)) 157.90/57.18
TOP(mark(nil)) → c23(TOP(ok(nil)), PROPER(nil)) 157.90/57.18
TOP(mark(s(z0))) → c23(TOP(s(proper(z0))), PROPER(s(z0))) 157.90/57.18
TOP(mark(cons(z0, z1))) → c23(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 157.90/57.18
TOP(mark(from(z0))) → c23(TOP(from(proper(z0))), PROPER(from(z0)))
S tuples:

ACTIVE(first(z0, z1)) → c3(FIRST(active(z0), z1), ACTIVE(z0)) 157.90/57.18
ACTIVE(first(z0, z1)) → c4(FIRST(z0, active(z1)), ACTIVE(z1)) 157.90/57.18
FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.90/57.18
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.90/57.18
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.90/57.18
S(mark(z0)) → c11(S(z0)) 157.90/57.18
S(ok(z0)) → c12(S(z0)) 157.90/57.18
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.90/57.18
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.90/57.18
FROM(mark(z0)) → c15(FROM(z0)) 157.90/57.18
FROM(ok(z0)) → c16(FROM(z0)) 157.90/57.18
TOP(ok(z0)) → c24(TOP(active(z0)), ACTIVE(z0)) 157.90/57.18
ACTIVE(from(z0)) → c2(S(z0)) 157.90/57.18
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 157.90/57.18
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 157.90/57.18
ACTIVE(s(first(0, z0))) → c5(S(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.18
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c5(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.18
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.18
ACTIVE(s(first(z0, z1))) → c5(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.18
ACTIVE(s(first(z0, z1))) → c5(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.18
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 157.90/57.18
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.18
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 157.90/57.18
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 157.90/57.18
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.18
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 157.90/57.18
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 157.90/57.18
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 157.90/57.18
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 157.90/57.18
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 157.90/57.18
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 157.90/57.18
ACTIVE(from(first(0, z0))) → c7(FROM(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.18
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c7(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.18
ACTIVE(from(from(z0))) → c7(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.18
ACTIVE(from(first(z0, z1))) → c7(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.18
ACTIVE(from(first(z0, z1))) → c7(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.18
ACTIVE(from(s(z0))) → c7(FROM(s(active(z0))), ACTIVE(s(z0))) 157.90/57.18
ACTIVE(from(cons(z0, z1))) → c7(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.18
ACTIVE(from(from(z0))) → c7(FROM(from(active(z0))), ACTIVE(from(z0))) 157.90/57.18
PROPER(first(x0, first(z0, z1))) → c17(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 157.90/57.18
PROPER(first(x0, s(z0))) → c17(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 157.90/57.18
PROPER(first(x0, cons(z0, z1))) → c17(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 157.90/57.18
PROPER(first(x0, from(z0))) → c17(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 157.90/57.18
PROPER(first(first(z0, z1), x1)) → c17(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 157.90/57.18
PROPER(first(s(z0), x1)) → c17(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 157.90/57.18
PROPER(first(cons(z0, z1), x1)) → c17(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 157.90/57.18
PROPER(first(from(z0), x1)) → c17(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 157.90/57.18
PROPER(first(x0, 0)) → c17(FIRST(proper(x0), ok(0)), PROPER(x0)) 157.90/57.18
PROPER(first(x0, nil)) → c17(FIRST(proper(x0), ok(nil)), PROPER(x0)) 157.90/57.18
PROPER(first(0, x1)) → c17(FIRST(ok(0), proper(x1)), PROPER(x1)) 157.90/57.18
PROPER(first(nil, x1)) → c17(FIRST(ok(nil), proper(x1)), PROPER(x1)) 157.90/57.18
PROPER(s(first(z0, z1))) → c20(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 157.90/57.18
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 157.90/57.18
PROPER(s(cons(z0, z1))) → c20(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 157.90/57.18
PROPER(s(from(z0))) → c20(S(from(proper(z0))), PROPER(from(z0))) 157.90/57.18
PROPER(s(0)) → c20(S(ok(0))) 157.90/57.18
PROPER(s(nil)) → c20(S(ok(nil))) 157.90/57.18
PROPER(cons(x0, first(z0, z1))) → c21(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 157.90/57.18
PROPER(cons(x0, s(z0))) → c21(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 157.90/57.18
PROPER(cons(x0, cons(z0, z1))) → c21(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 157.90/57.18
PROPER(cons(x0, from(z0))) → c21(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 157.90/57.18
PROPER(cons(first(z0, z1), x1)) → c21(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 157.90/57.18
PROPER(cons(s(z0), x1)) → c21(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 157.90/57.18
PROPER(cons(cons(z0, z1), x1)) → c21(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 157.90/57.18
PROPER(cons(from(z0), x1)) → c21(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 157.90/57.18
PROPER(cons(x0, 0)) → c21(CONS(proper(x0), ok(0)), PROPER(x0)) 157.90/57.18
PROPER(cons(x0, nil)) → c21(CONS(proper(x0), ok(nil)), PROPER(x0)) 157.90/57.18
PROPER(cons(0, x1)) → c21(CONS(ok(0), proper(x1)), PROPER(x1)) 157.90/57.18
PROPER(cons(nil, x1)) → c21(CONS(ok(nil), proper(x1)), PROPER(x1)) 157.90/57.18
PROPER(from(first(z0, z1))) → c22(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 157.90/57.18
PROPER(from(s(z0))) → c22(FROM(s(proper(z0))), PROPER(s(z0))) 157.90/57.18
PROPER(from(cons(z0, z1))) → c22(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 157.90/57.18
PROPER(from(from(z0))) → c22(FROM(from(proper(z0))), PROPER(from(z0))) 157.90/57.18
PROPER(from(0)) → c22(FROM(ok(0))) 157.90/57.18
PROPER(from(nil)) → c22(FROM(ok(nil)))
K tuples:

TOP(mark(z0)) → c23(TOP(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, first, s, cons, from, proper, top

Defined Pair Symbols:

ACTIVE, FIRST, S, CONS, FROM, TOP, PROPER

Compound Symbols:

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

157.90/57.18
157.90/57.18

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

Removed 2 trailing tuple parts
157.90/57.18
157.90/57.18

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil) 157.90/57.18
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 157.90/57.18
active(from(z0)) → mark(cons(z0, from(s(z0)))) 157.90/57.18
active(first(z0, z1)) → first(active(z0), z1) 157.90/57.18
active(first(z0, z1)) → first(z0, active(z1)) 157.90/57.18
active(s(z0)) → s(active(z0)) 157.90/57.18
active(cons(z0, z1)) → cons(active(z0), z1) 157.90/57.18
active(from(z0)) → from(active(z0)) 157.90/57.18
first(mark(z0), z1) → mark(first(z0, z1)) 157.90/57.18
first(z0, mark(z1)) → mark(first(z0, z1)) 157.90/57.18
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 157.90/57.18
s(mark(z0)) → mark(s(z0)) 157.90/57.18
s(ok(z0)) → ok(s(z0)) 157.90/57.18
cons(mark(z0), z1) → mark(cons(z0, z1)) 157.90/57.18
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 157.90/57.18
from(mark(z0)) → mark(from(z0)) 157.90/57.18
from(ok(z0)) → ok(from(z0)) 157.90/57.18
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 157.90/57.18
proper(0) → ok(0) 157.90/57.18
proper(nil) → ok(nil) 157.90/57.18
proper(s(z0)) → s(proper(z0)) 157.90/57.18
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 157.90/57.18
proper(from(z0)) → from(proper(z0)) 157.90/57.18
top(mark(z0)) → top(proper(z0)) 157.90/57.18
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(first(z0, z1)) → c3(FIRST(active(z0), z1), ACTIVE(z0)) 157.90/57.18
ACTIVE(first(z0, z1)) → c4(FIRST(z0, active(z1)), ACTIVE(z1)) 157.90/57.18
FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.90/57.18
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.90/57.18
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.90/57.18
S(mark(z0)) → c11(S(z0)) 157.90/57.18
S(ok(z0)) → c12(S(z0)) 157.90/57.18
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.90/57.18
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.90/57.18
FROM(mark(z0)) → c15(FROM(z0)) 157.90/57.18
FROM(ok(z0)) → c16(FROM(z0)) 157.90/57.18
TOP(ok(z0)) → c24(TOP(active(z0)), ACTIVE(z0)) 157.90/57.18
ACTIVE(from(z0)) → c2(S(z0)) 157.90/57.18
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 157.90/57.18
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 157.90/57.18
ACTIVE(s(first(0, z0))) → c5(S(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.18
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c5(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.18
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.18
ACTIVE(s(first(z0, z1))) → c5(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.18
ACTIVE(s(first(z0, z1))) → c5(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.18
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 157.90/57.18
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.18
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 157.90/57.18
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 157.90/57.18
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.18
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 157.90/57.18
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 157.90/57.18
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 157.90/57.18
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 157.90/57.18
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 157.90/57.18
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 157.90/57.18
ACTIVE(from(first(0, z0))) → c7(FROM(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.18
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c7(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.18
ACTIVE(from(from(z0))) → c7(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.18
ACTIVE(from(first(z0, z1))) → c7(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.18
ACTIVE(from(first(z0, z1))) → c7(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.18
ACTIVE(from(s(z0))) → c7(FROM(s(active(z0))), ACTIVE(s(z0))) 157.90/57.18
ACTIVE(from(cons(z0, z1))) → c7(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.18
ACTIVE(from(from(z0))) → c7(FROM(from(active(z0))), ACTIVE(from(z0))) 157.90/57.18
PROPER(first(x0, first(z0, z1))) → c17(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 157.90/57.18
PROPER(first(x0, s(z0))) → c17(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 157.90/57.18
PROPER(first(x0, cons(z0, z1))) → c17(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 157.90/57.18
PROPER(first(x0, from(z0))) → c17(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 157.90/57.18
PROPER(first(first(z0, z1), x1)) → c17(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 157.90/57.18
PROPER(first(s(z0), x1)) → c17(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 157.90/57.18
PROPER(first(cons(z0, z1), x1)) → c17(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 157.90/57.18
PROPER(first(from(z0), x1)) → c17(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 157.90/57.18
PROPER(first(x0, 0)) → c17(FIRST(proper(x0), ok(0)), PROPER(x0)) 157.90/57.18
PROPER(first(x0, nil)) → c17(FIRST(proper(x0), ok(nil)), PROPER(x0)) 157.90/57.18
PROPER(first(0, x1)) → c17(FIRST(ok(0), proper(x1)), PROPER(x1)) 157.90/57.18
PROPER(first(nil, x1)) → c17(FIRST(ok(nil), proper(x1)), PROPER(x1)) 157.90/57.18
PROPER(s(first(z0, z1))) → c20(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 157.90/57.18
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 157.90/57.18
PROPER(s(cons(z0, z1))) → c20(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 157.90/57.18
PROPER(s(from(z0))) → c20(S(from(proper(z0))), PROPER(from(z0))) 157.90/57.18
PROPER(s(0)) → c20(S(ok(0))) 157.90/57.18
PROPER(s(nil)) → c20(S(ok(nil))) 157.90/57.18
PROPER(cons(x0, first(z0, z1))) → c21(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 157.90/57.18
PROPER(cons(x0, s(z0))) → c21(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 157.90/57.18
PROPER(cons(x0, cons(z0, z1))) → c21(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 157.90/57.18
PROPER(cons(x0, from(z0))) → c21(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 157.90/57.18
PROPER(cons(first(z0, z1), x1)) → c21(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 157.90/57.18
PROPER(cons(s(z0), x1)) → c21(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 157.90/57.18
PROPER(cons(cons(z0, z1), x1)) → c21(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 157.90/57.19
PROPER(cons(from(z0), x1)) → c21(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 157.90/57.19
PROPER(cons(x0, 0)) → c21(CONS(proper(x0), ok(0)), PROPER(x0)) 157.90/57.19
PROPER(cons(x0, nil)) → c21(CONS(proper(x0), ok(nil)), PROPER(x0)) 157.90/57.19
PROPER(cons(0, x1)) → c21(CONS(ok(0), proper(x1)), PROPER(x1)) 157.90/57.19
PROPER(cons(nil, x1)) → c21(CONS(ok(nil), proper(x1)), PROPER(x1)) 157.90/57.19
PROPER(from(first(z0, z1))) → c22(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 157.90/57.19
PROPER(from(s(z0))) → c22(FROM(s(proper(z0))), PROPER(s(z0))) 157.90/57.19
PROPER(from(cons(z0, z1))) → c22(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 157.90/57.19
PROPER(from(from(z0))) → c22(FROM(from(proper(z0))), PROPER(from(z0))) 157.90/57.19
PROPER(from(0)) → c22(FROM(ok(0))) 157.90/57.19
PROPER(from(nil)) → c22(FROM(ok(nil))) 157.90/57.19
TOP(mark(first(z0, z1))) → c23(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 157.90/57.19
TOP(mark(s(z0))) → c23(TOP(s(proper(z0))), PROPER(s(z0))) 157.90/57.19
TOP(mark(cons(z0, z1))) → c23(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 157.90/57.19
TOP(mark(from(z0))) → c23(TOP(from(proper(z0))), PROPER(from(z0))) 157.90/57.19
TOP(mark(0)) → c23(TOP(ok(0))) 157.90/57.19
TOP(mark(nil)) → c23(TOP(ok(nil)))
S tuples:

ACTIVE(first(z0, z1)) → c3(FIRST(active(z0), z1), ACTIVE(z0)) 157.90/57.19
ACTIVE(first(z0, z1)) → c4(FIRST(z0, active(z1)), ACTIVE(z1)) 157.90/57.19
FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.90/57.19
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.90/57.19
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.90/57.19
S(mark(z0)) → c11(S(z0)) 157.90/57.19
S(ok(z0)) → c12(S(z0)) 157.90/57.19
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.90/57.19
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.90/57.19
FROM(mark(z0)) → c15(FROM(z0)) 157.90/57.19
FROM(ok(z0)) → c16(FROM(z0)) 157.90/57.19
TOP(ok(z0)) → c24(TOP(active(z0)), ACTIVE(z0)) 157.90/57.19
ACTIVE(from(z0)) → c2(S(z0)) 157.90/57.19
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 157.90/57.19
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 157.90/57.19
ACTIVE(s(first(0, z0))) → c5(S(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.19
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c5(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.19
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.19
ACTIVE(s(first(z0, z1))) → c5(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.19
ACTIVE(s(first(z0, z1))) → c5(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.19
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 157.90/57.19
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.19
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 157.90/57.19
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 157.90/57.19
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.19
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 157.90/57.19
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 157.90/57.19
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 157.90/57.19
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 157.90/57.19
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 157.90/57.19
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 157.90/57.19
ACTIVE(from(first(0, z0))) → c7(FROM(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.19
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c7(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.19
ACTIVE(from(from(z0))) → c7(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.19
ACTIVE(from(first(z0, z1))) → c7(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.19
ACTIVE(from(first(z0, z1))) → c7(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.19
ACTIVE(from(s(z0))) → c7(FROM(s(active(z0))), ACTIVE(s(z0))) 157.90/57.19
ACTIVE(from(cons(z0, z1))) → c7(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.19
ACTIVE(from(from(z0))) → c7(FROM(from(active(z0))), ACTIVE(from(z0))) 157.90/57.19
PROPER(first(x0, first(z0, z1))) → c17(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 157.90/57.19
PROPER(first(x0, s(z0))) → c17(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 157.90/57.19
PROPER(first(x0, cons(z0, z1))) → c17(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 157.90/57.19
PROPER(first(x0, from(z0))) → c17(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 157.90/57.19
PROPER(first(first(z0, z1), x1)) → c17(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 157.90/57.19
PROPER(first(s(z0), x1)) → c17(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 157.90/57.19
PROPER(first(cons(z0, z1), x1)) → c17(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 157.90/57.19
PROPER(first(from(z0), x1)) → c17(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 157.90/57.19
PROPER(first(x0, 0)) → c17(FIRST(proper(x0), ok(0)), PROPER(x0)) 157.90/57.19
PROPER(first(x0, nil)) → c17(FIRST(proper(x0), ok(nil)), PROPER(x0)) 157.90/57.19
PROPER(first(0, x1)) → c17(FIRST(ok(0), proper(x1)), PROPER(x1)) 157.90/57.19
PROPER(first(nil, x1)) → c17(FIRST(ok(nil), proper(x1)), PROPER(x1)) 157.90/57.19
PROPER(s(first(z0, z1))) → c20(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 157.90/57.19
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 157.90/57.19
PROPER(s(cons(z0, z1))) → c20(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 157.90/57.19
PROPER(s(from(z0))) → c20(S(from(proper(z0))), PROPER(from(z0))) 157.90/57.19
PROPER(s(0)) → c20(S(ok(0))) 157.90/57.19
PROPER(s(nil)) → c20(S(ok(nil))) 157.90/57.19
PROPER(cons(x0, first(z0, z1))) → c21(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 157.90/57.19
PROPER(cons(x0, s(z0))) → c21(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 157.90/57.19
PROPER(cons(x0, cons(z0, z1))) → c21(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 157.90/57.19
PROPER(cons(x0, from(z0))) → c21(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 157.90/57.19
PROPER(cons(first(z0, z1), x1)) → c21(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 157.90/57.19
PROPER(cons(s(z0), x1)) → c21(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 157.90/57.19
PROPER(cons(cons(z0, z1), x1)) → c21(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 157.90/57.19
PROPER(cons(from(z0), x1)) → c21(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 157.90/57.19
PROPER(cons(x0, 0)) → c21(CONS(proper(x0), ok(0)), PROPER(x0)) 157.90/57.19
PROPER(cons(x0, nil)) → c21(CONS(proper(x0), ok(nil)), PROPER(x0)) 157.90/57.19
PROPER(cons(0, x1)) → c21(CONS(ok(0), proper(x1)), PROPER(x1)) 157.90/57.19
PROPER(cons(nil, x1)) → c21(CONS(ok(nil), proper(x1)), PROPER(x1)) 157.90/57.19
PROPER(from(first(z0, z1))) → c22(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 157.90/57.19
PROPER(from(s(z0))) → c22(FROM(s(proper(z0))), PROPER(s(z0))) 157.90/57.19
PROPER(from(cons(z0, z1))) → c22(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 157.90/57.19
PROPER(from(from(z0))) → c22(FROM(from(proper(z0))), PROPER(from(z0))) 157.90/57.19
PROPER(from(0)) → c22(FROM(ok(0))) 157.90/57.19
PROPER(from(nil)) → c22(FROM(ok(nil)))
K tuples:

TOP(mark(z0)) → c23(TOP(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, first, s, cons, from, proper, top

Defined Pair Symbols:

ACTIVE, FIRST, S, CONS, FROM, TOP, PROPER

Compound Symbols:

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

157.90/57.19
157.90/57.19

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

Use narrowing to replace TOP(ok(z0)) → c24(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(first(0, z0))) → c24(TOP(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.19
TOP(ok(first(s(z0), cons(z1, z2)))) → c24(TOP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.19
TOP(ok(from(z0))) → c24(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.19
TOP(ok(first(z0, z1))) → c24(TOP(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.19
TOP(ok(first(z0, z1))) → c24(TOP(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.19
TOP(ok(s(z0))) → c24(TOP(s(active(z0))), ACTIVE(s(z0))) 157.90/57.19
TOP(ok(cons(z0, z1))) → c24(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.19
TOP(ok(from(z0))) → c24(TOP(from(active(z0))), ACTIVE(from(z0)))
157.90/57.19
157.90/57.19

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil) 157.90/57.19
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 157.90/57.19
active(from(z0)) → mark(cons(z0, from(s(z0)))) 157.90/57.19
active(first(z0, z1)) → first(active(z0), z1) 157.90/57.19
active(first(z0, z1)) → first(z0, active(z1)) 157.90/57.19
active(s(z0)) → s(active(z0)) 157.90/57.19
active(cons(z0, z1)) → cons(active(z0), z1) 157.90/57.19
active(from(z0)) → from(active(z0)) 157.90/57.19
first(mark(z0), z1) → mark(first(z0, z1)) 157.90/57.19
first(z0, mark(z1)) → mark(first(z0, z1)) 157.90/57.19
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 157.90/57.19
s(mark(z0)) → mark(s(z0)) 157.90/57.19
s(ok(z0)) → ok(s(z0)) 157.90/57.19
cons(mark(z0), z1) → mark(cons(z0, z1)) 157.90/57.19
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 157.90/57.19
from(mark(z0)) → mark(from(z0)) 157.90/57.19
from(ok(z0)) → ok(from(z0)) 157.90/57.19
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 157.90/57.19
proper(0) → ok(0) 157.90/57.19
proper(nil) → ok(nil) 157.90/57.19
proper(s(z0)) → s(proper(z0)) 157.90/57.19
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 157.90/57.19
proper(from(z0)) → from(proper(z0)) 157.90/57.19
top(mark(z0)) → top(proper(z0)) 157.90/57.19
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(first(z0, z1)) → c3(FIRST(active(z0), z1), ACTIVE(z0)) 157.90/57.19
ACTIVE(first(z0, z1)) → c4(FIRST(z0, active(z1)), ACTIVE(z1)) 157.90/57.19
FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.90/57.19
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.90/57.19
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.90/57.19
S(mark(z0)) → c11(S(z0)) 157.90/57.19
S(ok(z0)) → c12(S(z0)) 157.90/57.19
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.90/57.19
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.90/57.19
FROM(mark(z0)) → c15(FROM(z0)) 157.90/57.19
FROM(ok(z0)) → c16(FROM(z0)) 157.90/57.19
ACTIVE(from(z0)) → c2(S(z0)) 157.90/57.19
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 157.90/57.19
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 157.90/57.19
ACTIVE(s(first(0, z0))) → c5(S(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.19
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c5(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.19
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.19
ACTIVE(s(first(z0, z1))) → c5(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.19
ACTIVE(s(first(z0, z1))) → c5(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.19
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 157.90/57.19
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.19
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 157.90/57.19
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 157.90/57.19
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.19
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 157.90/57.19
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 157.90/57.19
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 157.90/57.19
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 157.90/57.19
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 157.90/57.19
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 157.90/57.19
ACTIVE(from(first(0, z0))) → c7(FROM(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.19
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c7(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.19
ACTIVE(from(from(z0))) → c7(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.19
ACTIVE(from(first(z0, z1))) → c7(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.19
ACTIVE(from(first(z0, z1))) → c7(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.19
ACTIVE(from(s(z0))) → c7(FROM(s(active(z0))), ACTIVE(s(z0))) 157.90/57.19
ACTIVE(from(cons(z0, z1))) → c7(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.19
ACTIVE(from(from(z0))) → c7(FROM(from(active(z0))), ACTIVE(from(z0))) 157.90/57.19
PROPER(first(x0, first(z0, z1))) → c17(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 157.90/57.19
PROPER(first(x0, s(z0))) → c17(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 157.90/57.19
PROPER(first(x0, cons(z0, z1))) → c17(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 157.90/57.19
PROPER(first(x0, from(z0))) → c17(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 157.90/57.19
PROPER(first(first(z0, z1), x1)) → c17(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 157.90/57.19
PROPER(first(s(z0), x1)) → c17(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 157.90/57.19
PROPER(first(cons(z0, z1), x1)) → c17(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 157.90/57.19
PROPER(first(from(z0), x1)) → c17(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 157.90/57.19
PROPER(first(x0, 0)) → c17(FIRST(proper(x0), ok(0)), PROPER(x0)) 157.90/57.19
PROPER(first(x0, nil)) → c17(FIRST(proper(x0), ok(nil)), PROPER(x0)) 157.90/57.19
PROPER(first(0, x1)) → c17(FIRST(ok(0), proper(x1)), PROPER(x1)) 157.90/57.19
PROPER(first(nil, x1)) → c17(FIRST(ok(nil), proper(x1)), PROPER(x1)) 157.90/57.19
PROPER(s(first(z0, z1))) → c20(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 157.90/57.19
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 157.90/57.19
PROPER(s(cons(z0, z1))) → c20(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 157.90/57.19
PROPER(s(from(z0))) → c20(S(from(proper(z0))), PROPER(from(z0))) 157.90/57.19
PROPER(s(0)) → c20(S(ok(0))) 157.90/57.19
PROPER(s(nil)) → c20(S(ok(nil))) 157.90/57.19
PROPER(cons(x0, first(z0, z1))) → c21(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 157.90/57.19
PROPER(cons(x0, s(z0))) → c21(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 157.90/57.19
PROPER(cons(x0, cons(z0, z1))) → c21(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 157.90/57.19
PROPER(cons(x0, from(z0))) → c21(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 157.90/57.19
PROPER(cons(first(z0, z1), x1)) → c21(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 157.90/57.19
PROPER(cons(s(z0), x1)) → c21(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 157.90/57.19
PROPER(cons(cons(z0, z1), x1)) → c21(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 157.90/57.19
PROPER(cons(from(z0), x1)) → c21(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 157.90/57.19
PROPER(cons(x0, 0)) → c21(CONS(proper(x0), ok(0)), PROPER(x0)) 157.90/57.19
PROPER(cons(x0, nil)) → c21(CONS(proper(x0), ok(nil)), PROPER(x0)) 157.90/57.19
PROPER(cons(0, x1)) → c21(CONS(ok(0), proper(x1)), PROPER(x1)) 157.90/57.19
PROPER(cons(nil, x1)) → c21(CONS(ok(nil), proper(x1)), PROPER(x1)) 157.90/57.19
PROPER(from(first(z0, z1))) → c22(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 157.90/57.19
PROPER(from(s(z0))) → c22(FROM(s(proper(z0))), PROPER(s(z0))) 157.90/57.19
PROPER(from(cons(z0, z1))) → c22(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 157.90/57.19
PROPER(from(from(z0))) → c22(FROM(from(proper(z0))), PROPER(from(z0))) 157.90/57.19
PROPER(from(0)) → c22(FROM(ok(0))) 157.90/57.19
PROPER(from(nil)) → c22(FROM(ok(nil))) 157.90/57.19
TOP(mark(first(z0, z1))) → c23(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 157.90/57.19
TOP(mark(s(z0))) → c23(TOP(s(proper(z0))), PROPER(s(z0))) 157.90/57.19
TOP(mark(cons(z0, z1))) → c23(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 157.90/57.19
TOP(mark(from(z0))) → c23(TOP(from(proper(z0))), PROPER(from(z0))) 157.90/57.19
TOP(mark(0)) → c23(TOP(ok(0))) 157.90/57.19
TOP(mark(nil)) → c23(TOP(ok(nil))) 157.90/57.19
TOP(ok(first(0, z0))) → c24(TOP(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.19
TOP(ok(first(s(z0), cons(z1, z2)))) → c24(TOP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.19
TOP(ok(from(z0))) → c24(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.19
TOP(ok(first(z0, z1))) → c24(TOP(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.19
TOP(ok(first(z0, z1))) → c24(TOP(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.19
TOP(ok(s(z0))) → c24(TOP(s(active(z0))), ACTIVE(s(z0))) 157.90/57.19
TOP(ok(cons(z0, z1))) → c24(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.19
TOP(ok(from(z0))) → c24(TOP(from(active(z0))), ACTIVE(from(z0)))
S tuples:

ACTIVE(first(z0, z1)) → c3(FIRST(active(z0), z1), ACTIVE(z0)) 157.90/57.19
ACTIVE(first(z0, z1)) → c4(FIRST(z0, active(z1)), ACTIVE(z1)) 157.90/57.19
FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.90/57.19
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.90/57.19
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.90/57.19
S(mark(z0)) → c11(S(z0)) 157.90/57.19
S(ok(z0)) → c12(S(z0)) 157.90/57.19
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.90/57.19
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.90/57.19
FROM(mark(z0)) → c15(FROM(z0)) 157.90/57.19
FROM(ok(z0)) → c16(FROM(z0)) 157.90/57.19
ACTIVE(from(z0)) → c2(S(z0)) 157.90/57.19
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 157.90/57.19
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 157.90/57.19
ACTIVE(s(first(0, z0))) → c5(S(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.19
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c5(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.19
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.19
ACTIVE(s(first(z0, z1))) → c5(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.19
ACTIVE(s(first(z0, z1))) → c5(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.19
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 157.90/57.19
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.19
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 157.90/57.19
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 157.90/57.19
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.19
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 157.90/57.19
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 157.90/57.19
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 157.90/57.19
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 157.90/57.19
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 157.90/57.19
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 157.90/57.19
ACTIVE(from(first(0, z0))) → c7(FROM(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.19
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c7(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.19
ACTIVE(from(from(z0))) → c7(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.19
ACTIVE(from(first(z0, z1))) → c7(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.19
ACTIVE(from(first(z0, z1))) → c7(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.19
ACTIVE(from(s(z0))) → c7(FROM(s(active(z0))), ACTIVE(s(z0))) 157.90/57.19
ACTIVE(from(cons(z0, z1))) → c7(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.19
ACTIVE(from(from(z0))) → c7(FROM(from(active(z0))), ACTIVE(from(z0))) 157.90/57.19
PROPER(first(x0, first(z0, z1))) → c17(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 157.90/57.19
PROPER(first(x0, s(z0))) → c17(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 157.90/57.19
PROPER(first(x0, cons(z0, z1))) → c17(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 157.90/57.19
PROPER(first(x0, from(z0))) → c17(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 157.90/57.19
PROPER(first(first(z0, z1), x1)) → c17(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 157.90/57.19
PROPER(first(s(z0), x1)) → c17(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 157.90/57.19
PROPER(first(cons(z0, z1), x1)) → c17(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 157.90/57.19
PROPER(first(from(z0), x1)) → c17(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 157.90/57.19
PROPER(first(x0, 0)) → c17(FIRST(proper(x0), ok(0)), PROPER(x0)) 157.90/57.19
PROPER(first(x0, nil)) → c17(FIRST(proper(x0), ok(nil)), PROPER(x0)) 157.90/57.19
PROPER(first(0, x1)) → c17(FIRST(ok(0), proper(x1)), PROPER(x1)) 157.90/57.19
PROPER(first(nil, x1)) → c17(FIRST(ok(nil), proper(x1)), PROPER(x1)) 157.90/57.19
PROPER(s(first(z0, z1))) → c20(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 157.90/57.19
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 157.90/57.19
PROPER(s(cons(z0, z1))) → c20(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 157.90/57.19
PROPER(s(from(z0))) → c20(S(from(proper(z0))), PROPER(from(z0))) 157.90/57.19
PROPER(s(0)) → c20(S(ok(0))) 157.90/57.19
PROPER(s(nil)) → c20(S(ok(nil))) 157.90/57.19
PROPER(cons(x0, first(z0, z1))) → c21(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 157.90/57.19
PROPER(cons(x0, s(z0))) → c21(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 157.90/57.19
PROPER(cons(x0, cons(z0, z1))) → c21(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 157.90/57.19
PROPER(cons(x0, from(z0))) → c21(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 157.90/57.19
PROPER(cons(first(z0, z1), x1)) → c21(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 157.90/57.19
PROPER(cons(s(z0), x1)) → c21(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 157.90/57.19
PROPER(cons(cons(z0, z1), x1)) → c21(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 157.90/57.19
PROPER(cons(from(z0), x1)) → c21(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 157.90/57.19
PROPER(cons(x0, 0)) → c21(CONS(proper(x0), ok(0)), PROPER(x0)) 157.90/57.19
PROPER(cons(x0, nil)) → c21(CONS(proper(x0), ok(nil)), PROPER(x0)) 157.90/57.19
PROPER(cons(0, x1)) → c21(CONS(ok(0), proper(x1)), PROPER(x1)) 157.90/57.19
PROPER(cons(nil, x1)) → c21(CONS(ok(nil), proper(x1)), PROPER(x1)) 157.90/57.19
PROPER(from(first(z0, z1))) → c22(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 157.90/57.19
PROPER(from(s(z0))) → c22(FROM(s(proper(z0))), PROPER(s(z0))) 157.90/57.19
PROPER(from(cons(z0, z1))) → c22(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 157.90/57.19
PROPER(from(from(z0))) → c22(FROM(from(proper(z0))), PROPER(from(z0))) 157.90/57.19
PROPER(from(0)) → c22(FROM(ok(0))) 157.90/57.19
PROPER(from(nil)) → c22(FROM(ok(nil))) 157.90/57.19
TOP(ok(first(0, z0))) → c24(TOP(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.19
TOP(ok(first(s(z0), cons(z1, z2)))) → c24(TOP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.19
TOP(ok(from(z0))) → c24(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.19
TOP(ok(first(z0, z1))) → c24(TOP(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.19
TOP(ok(first(z0, z1))) → c24(TOP(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.19
TOP(ok(s(z0))) → c24(TOP(s(active(z0))), ACTIVE(s(z0))) 157.90/57.19
TOP(ok(cons(z0, z1))) → c24(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.19
TOP(ok(from(z0))) → c24(TOP(from(active(z0))), ACTIVE(from(z0)))
K tuples:

TOP(mark(z0)) → c23(TOP(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, first, s, cons, from, proper, top

Defined Pair Symbols:

ACTIVE, FIRST, S, CONS, FROM, PROPER, TOP

Compound Symbols:

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

157.90/57.19
157.90/57.19

(37) CdtUnreachableProof (EQUIVALENT transformation)

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

ACTIVE(first(z0, z1)) → c3(FIRST(active(z0), z1), ACTIVE(z0)) 157.90/57.19
ACTIVE(first(z0, z1)) → c4(FIRST(z0, active(z1)), ACTIVE(z1)) 157.90/57.19
ACTIVE(from(z0)) → c2(S(z0)) 157.90/57.19
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2))) 157.90/57.19
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2)) 157.90/57.19
ACTIVE(s(first(0, z0))) → c5(S(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.19
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c5(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.19
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.19
ACTIVE(s(first(z0, z1))) → c5(S(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.19
ACTIVE(s(first(z0, z1))) → c5(S(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.19
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 157.90/57.19
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.19
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 157.90/57.19
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0))) 157.90/57.19
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.19
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 157.90/57.19
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1))) 157.90/57.19
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1))) 157.90/57.19
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 157.90/57.19
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 157.90/57.19
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 157.90/57.19
ACTIVE(from(first(0, z0))) → c7(FROM(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.19
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c7(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.19
ACTIVE(from(from(z0))) → c7(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.19
ACTIVE(from(first(z0, z1))) → c7(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.19
ACTIVE(from(first(z0, z1))) → c7(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.19
ACTIVE(from(s(z0))) → c7(FROM(s(active(z0))), ACTIVE(s(z0))) 157.90/57.19
ACTIVE(from(cons(z0, z1))) → c7(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.19
ACTIVE(from(from(z0))) → c7(FROM(from(active(z0))), ACTIVE(from(z0))) 157.90/57.19
PROPER(first(x0, first(z0, z1))) → c17(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 157.90/57.19
PROPER(first(x0, s(z0))) → c17(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 157.90/57.19
PROPER(first(x0, cons(z0, z1))) → c17(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 157.90/57.19
PROPER(first(x0, from(z0))) → c17(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 157.90/57.19
PROPER(first(first(z0, z1), x1)) → c17(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 157.90/57.19
PROPER(first(s(z0), x1)) → c17(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 157.90/57.19
PROPER(first(cons(z0, z1), x1)) → c17(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 157.90/57.19
PROPER(first(from(z0), x1)) → c17(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 157.90/57.19
PROPER(first(x0, 0)) → c17(FIRST(proper(x0), ok(0)), PROPER(x0)) 157.90/57.19
PROPER(first(x0, nil)) → c17(FIRST(proper(x0), ok(nil)), PROPER(x0)) 157.90/57.19
PROPER(first(0, x1)) → c17(FIRST(ok(0), proper(x1)), PROPER(x1)) 157.90/57.19
PROPER(first(nil, x1)) → c17(FIRST(ok(nil), proper(x1)), PROPER(x1)) 157.90/57.19
PROPER(s(first(z0, z1))) → c20(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 157.90/57.19
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 157.90/57.19
PROPER(s(cons(z0, z1))) → c20(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 157.90/57.19
PROPER(s(from(z0))) → c20(S(from(proper(z0))), PROPER(from(z0))) 157.90/57.19
PROPER(s(0)) → c20(S(ok(0))) 157.90/57.19
PROPER(s(nil)) → c20(S(ok(nil))) 157.90/57.19
PROPER(cons(x0, first(z0, z1))) → c21(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1))) 157.90/57.19
PROPER(cons(x0, s(z0))) → c21(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 157.90/57.19
PROPER(cons(x0, cons(z0, z1))) → c21(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 157.90/57.19
PROPER(cons(x0, from(z0))) → c21(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 157.90/57.19
PROPER(cons(first(z0, z1), x1)) → c21(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1)) 157.90/57.19
PROPER(cons(s(z0), x1)) → c21(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 157.90/57.19
PROPER(cons(cons(z0, z1), x1)) → c21(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 157.90/57.19
PROPER(cons(from(z0), x1)) → c21(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 157.90/57.19
PROPER(cons(x0, 0)) → c21(CONS(proper(x0), ok(0)), PROPER(x0)) 157.90/57.19
PROPER(cons(x0, nil)) → c21(CONS(proper(x0), ok(nil)), PROPER(x0)) 157.90/57.19
PROPER(cons(0, x1)) → c21(CONS(ok(0), proper(x1)), PROPER(x1)) 157.90/57.19
PROPER(cons(nil, x1)) → c21(CONS(ok(nil), proper(x1)), PROPER(x1)) 157.90/57.19
PROPER(from(first(z0, z1))) → c22(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 157.90/57.19
PROPER(from(s(z0))) → c22(FROM(s(proper(z0))), PROPER(s(z0))) 157.90/57.19
PROPER(from(cons(z0, z1))) → c22(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 157.90/57.19
PROPER(from(from(z0))) → c22(FROM(from(proper(z0))), PROPER(from(z0))) 157.90/57.19
PROPER(from(0)) → c22(FROM(ok(0))) 157.90/57.19
PROPER(from(nil)) → c22(FROM(ok(nil))) 157.90/57.19
TOP(mark(first(z0, z1))) → c23(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1))) 157.90/57.19
TOP(mark(s(z0))) → c23(TOP(s(proper(z0))), PROPER(s(z0))) 157.90/57.19
TOP(mark(cons(z0, z1))) → c23(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 157.90/57.19
TOP(mark(from(z0))) → c23(TOP(from(proper(z0))), PROPER(from(z0))) 157.90/57.19
TOP(ok(first(0, z0))) → c24(TOP(mark(nil)), ACTIVE(first(0, z0))) 157.90/57.19
TOP(ok(first(s(z0), cons(z1, z2)))) → c24(TOP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2)))) 157.90/57.19
TOP(ok(from(z0))) → c24(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 157.90/57.19
TOP(ok(first(z0, z1))) → c24(TOP(first(active(z0), z1)), ACTIVE(first(z0, z1))) 157.90/57.19
TOP(ok(first(z0, z1))) → c24(TOP(first(z0, active(z1))), ACTIVE(first(z0, z1))) 157.90/57.19
TOP(ok(s(z0))) → c24(TOP(s(active(z0))), ACTIVE(s(z0))) 157.90/57.19
TOP(ok(cons(z0, z1))) → c24(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 157.90/57.19
TOP(ok(from(z0))) → c24(TOP(from(active(z0))), ACTIVE(from(z0)))
157.90/57.19
157.90/57.19

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil) 157.90/57.19
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 157.90/57.19
active(from(z0)) → mark(cons(z0, from(s(z0)))) 157.90/57.19
active(first(z0, z1)) → first(active(z0), z1) 157.90/57.19
active(first(z0, z1)) → first(z0, active(z1)) 157.90/57.19
active(s(z0)) → s(active(z0)) 157.90/57.19
active(cons(z0, z1)) → cons(active(z0), z1) 157.90/57.19
active(from(z0)) → from(active(z0)) 157.90/57.19
first(mark(z0), z1) → mark(first(z0, z1)) 157.90/57.19
first(z0, mark(z1)) → mark(first(z0, z1)) 157.90/57.19
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 157.90/57.19
s(mark(z0)) → mark(s(z0)) 157.90/57.19
s(ok(z0)) → ok(s(z0)) 157.90/57.19
cons(mark(z0), z1) → mark(cons(z0, z1)) 157.90/57.19
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 157.90/57.19
from(mark(z0)) → mark(from(z0)) 157.90/57.19
from(ok(z0)) → ok(from(z0)) 157.90/57.19
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 157.90/57.19
proper(0) → ok(0) 157.90/57.19
proper(nil) → ok(nil) 157.90/57.19
proper(s(z0)) → s(proper(z0)) 157.90/57.19
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 157.90/57.19
proper(from(z0)) → from(proper(z0)) 157.90/57.19
top(mark(z0)) → top(proper(z0)) 157.90/57.19
top(ok(z0)) → top(active(z0))
Tuples:

FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.90/57.19
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.90/57.19
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.90/57.19
S(mark(z0)) → c11(S(z0)) 157.90/57.19
S(ok(z0)) → c12(S(z0)) 157.90/57.19
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.90/57.19
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.90/57.19
FROM(mark(z0)) → c15(FROM(z0)) 157.90/57.19
FROM(ok(z0)) → c16(FROM(z0)) 157.90/57.19
TOP(mark(0)) → c23(TOP(ok(0))) 157.90/57.19
TOP(mark(nil)) → c23(TOP(ok(nil)))
S tuples:

FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.90/57.19
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.90/57.19
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.90/57.19
S(mark(z0)) → c11(S(z0)) 157.90/57.19
S(ok(z0)) → c12(S(z0)) 157.90/57.19
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.90/57.19
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.90/57.19
FROM(mark(z0)) → c15(FROM(z0)) 157.90/57.19
FROM(ok(z0)) → c16(FROM(z0))
K tuples:none
Defined Rule Symbols:

active, first, s, cons, from, proper, top

Defined Pair Symbols:

FIRST, S, CONS, FROM, TOP

Compound Symbols:

c8, c9, c10, c11, c12, c13, c14, c15, c16, c23

157.90/57.19
157.90/57.19

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

Removed 2 trailing tuple parts
157.90/57.19
157.90/57.19

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil) 157.90/57.19
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 157.90/57.19
active(from(z0)) → mark(cons(z0, from(s(z0)))) 157.90/57.19
active(first(z0, z1)) → first(active(z0), z1) 157.90/57.19
active(first(z0, z1)) → first(z0, active(z1)) 157.90/57.19
active(s(z0)) → s(active(z0)) 157.90/57.19
active(cons(z0, z1)) → cons(active(z0), z1) 157.90/57.19
active(from(z0)) → from(active(z0)) 157.90/57.19
first(mark(z0), z1) → mark(first(z0, z1)) 157.90/57.19
first(z0, mark(z1)) → mark(first(z0, z1)) 157.90/57.19
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 157.90/57.19
s(mark(z0)) → mark(s(z0)) 157.90/57.19
s(ok(z0)) → ok(s(z0)) 157.90/57.19
cons(mark(z0), z1) → mark(cons(z0, z1)) 157.90/57.19
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 157.90/57.19
from(mark(z0)) → mark(from(z0)) 157.90/57.19
from(ok(z0)) → ok(from(z0)) 157.90/57.19
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 157.90/57.19
proper(0) → ok(0) 157.90/57.19
proper(nil) → ok(nil) 157.90/57.19
proper(s(z0)) → s(proper(z0)) 157.90/57.19
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 157.90/57.19
proper(from(z0)) → from(proper(z0)) 157.90/57.19
top(mark(z0)) → top(proper(z0)) 157.90/57.19
top(ok(z0)) → top(active(z0))
Tuples:

FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.90/57.19
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.90/57.19
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.90/57.19
S(mark(z0)) → c11(S(z0)) 157.90/57.19
S(ok(z0)) → c12(S(z0)) 157.90/57.19
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.90/57.19
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.90/57.19
FROM(mark(z0)) → c15(FROM(z0)) 157.90/57.19
FROM(ok(z0)) → c16(FROM(z0)) 157.90/57.19
TOP(mark(0)) → c23 157.90/57.19
TOP(mark(nil)) → c23
S tuples:

FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.90/57.19
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.90/57.19
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.90/57.19
S(mark(z0)) → c11(S(z0)) 157.90/57.19
S(ok(z0)) → c12(S(z0)) 157.90/57.19
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.90/57.19
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.90/57.19
FROM(mark(z0)) → c15(FROM(z0)) 157.90/57.19
FROM(ok(z0)) → c16(FROM(z0))
K tuples:none
Defined Rule Symbols:

active, first, s, cons, from, proper, top

Defined Pair Symbols:

FIRST, S, CONS, FROM, TOP

Compound Symbols:

c8, c9, c10, c11, c12, c13, c14, c15, c16, c23

157.90/57.19
157.90/57.19

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

Removed 2 trailing nodes:

TOP(mark(nil)) → c23 157.90/57.19
TOP(mark(0)) → c23
157.90/57.19
157.90/57.19

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil) 157.90/57.19
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 157.90/57.19
active(from(z0)) → mark(cons(z0, from(s(z0)))) 157.90/57.19
active(first(z0, z1)) → first(active(z0), z1) 157.90/57.19
active(first(z0, z1)) → first(z0, active(z1)) 157.90/57.19
active(s(z0)) → s(active(z0)) 157.90/57.19
active(cons(z0, z1)) → cons(active(z0), z1) 157.90/57.19
active(from(z0)) → from(active(z0)) 157.90/57.19
first(mark(z0), z1) → mark(first(z0, z1)) 157.90/57.19
first(z0, mark(z1)) → mark(first(z0, z1)) 157.90/57.19
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 157.90/57.19
s(mark(z0)) → mark(s(z0)) 157.90/57.19
s(ok(z0)) → ok(s(z0)) 157.90/57.19
cons(mark(z0), z1) → mark(cons(z0, z1)) 157.90/57.19
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 157.90/57.19
from(mark(z0)) → mark(from(z0)) 157.90/57.19
from(ok(z0)) → ok(from(z0)) 157.90/57.19
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 157.90/57.19
proper(0) → ok(0) 157.90/57.19
proper(nil) → ok(nil) 157.90/57.19
proper(s(z0)) → s(proper(z0)) 157.90/57.19
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 157.90/57.19
proper(from(z0)) → from(proper(z0)) 157.90/57.19
top(mark(z0)) → top(proper(z0)) 157.90/57.19
top(ok(z0)) → top(active(z0))
Tuples:

FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.90/57.19
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.90/57.19
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.90/57.19
S(mark(z0)) → c11(S(z0)) 157.90/57.19
S(ok(z0)) → c12(S(z0)) 157.90/57.19
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.90/57.19
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.90/57.19
FROM(mark(z0)) → c15(FROM(z0)) 157.90/57.19
FROM(ok(z0)) → c16(FROM(z0))
S tuples:

FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.90/57.19
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.90/57.19
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.90/57.19
S(mark(z0)) → c11(S(z0)) 157.90/57.19
S(ok(z0)) → c12(S(z0)) 157.90/57.19
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.90/57.19
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.90/57.19
FROM(mark(z0)) → c15(FROM(z0)) 157.90/57.19
FROM(ok(z0)) → c16(FROM(z0))
K tuples:none
Defined Rule Symbols:

active, first, s, cons, from, proper, top

Defined Pair Symbols:

FIRST, S, CONS, FROM

Compound Symbols:

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

157.90/57.19
157.90/57.19

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

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

S(mark(z0)) → c11(S(z0)) 157.90/57.19
S(ok(z0)) → c12(S(z0)) 157.90/57.19
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.90/57.19
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.90/57.19
FROM(mark(z0)) → c15(FROM(z0)) 157.90/57.19
FROM(ok(z0)) → c16(FROM(z0))
We considered the (Usable) Rules:none
And the Tuples:

FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.90/57.19
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.90/57.19
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.90/57.19
S(mark(z0)) → c11(S(z0)) 157.90/57.19
S(ok(z0)) → c12(S(z0)) 157.90/57.19
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.90/57.19
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.90/57.19
FROM(mark(z0)) → c15(FROM(z0)) 157.90/57.19
FROM(ok(z0)) → c16(FROM(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 157.90/57.19

POL(CONS(x1, x2)) = [2]x1    157.90/57.19
POL(FIRST(x1, x2)) = 0    157.90/57.19
POL(FROM(x1)) = x1    157.90/57.19
POL(S(x1)) = x1    157.90/57.19
POL(c10(x1)) = x1    157.90/57.19
POL(c11(x1)) = x1    157.90/57.19
POL(c12(x1)) = x1    157.90/57.19
POL(c13(x1)) = x1    157.90/57.19
POL(c14(x1)) = x1    157.90/57.19
POL(c15(x1)) = x1    157.90/57.19
POL(c16(x1)) = x1    157.90/57.19
POL(c8(x1)) = x1    157.90/57.19
POL(c9(x1)) = x1    157.90/57.19
POL(mark(x1)) = [1] + x1    157.90/57.19
POL(ok(x1)) = [1] + x1   
157.90/57.19
157.90/57.19

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil) 157.90/57.19
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 157.90/57.19
active(from(z0)) → mark(cons(z0, from(s(z0)))) 157.90/57.19
active(first(z0, z1)) → first(active(z0), z1) 157.90/57.19
active(first(z0, z1)) → first(z0, active(z1)) 157.90/57.19
active(s(z0)) → s(active(z0)) 157.90/57.19
active(cons(z0, z1)) → cons(active(z0), z1) 157.90/57.19
active(from(z0)) → from(active(z0)) 157.90/57.19
first(mark(z0), z1) → mark(first(z0, z1)) 157.90/57.19
first(z0, mark(z1)) → mark(first(z0, z1)) 157.90/57.19
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 157.90/57.19
s(mark(z0)) → mark(s(z0)) 157.90/57.19
s(ok(z0)) → ok(s(z0)) 157.90/57.19
cons(mark(z0), z1) → mark(cons(z0, z1)) 157.90/57.19
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 157.90/57.19
from(mark(z0)) → mark(from(z0)) 157.90/57.19
from(ok(z0)) → ok(from(z0)) 157.90/57.19
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 157.90/57.19
proper(0) → ok(0) 157.90/57.19
proper(nil) → ok(nil) 157.90/57.19
proper(s(z0)) → s(proper(z0)) 157.90/57.19
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 157.90/57.19
proper(from(z0)) → from(proper(z0)) 157.90/57.19
top(mark(z0)) → top(proper(z0)) 157.90/57.19
top(ok(z0)) → top(active(z0))
Tuples:

FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.90/57.19
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.90/57.19
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.90/57.19
S(mark(z0)) → c11(S(z0)) 157.90/57.19
S(ok(z0)) → c12(S(z0)) 157.90/57.19
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.90/57.19
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.90/57.19
FROM(mark(z0)) → c15(FROM(z0)) 157.90/57.19
FROM(ok(z0)) → c16(FROM(z0))
S tuples:

FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.90/57.19
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.90/57.19
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1))
K tuples:

S(mark(z0)) → c11(S(z0)) 157.90/57.19
S(ok(z0)) → c12(S(z0)) 157.90/57.19
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.90/57.19
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.90/57.19
FROM(mark(z0)) → c15(FROM(z0)) 157.90/57.19
FROM(ok(z0)) → c16(FROM(z0))
Defined Rule Symbols:

active, first, s, cons, from, proper, top

Defined Pair Symbols:

FIRST, S, CONS, FROM

Compound Symbols:

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

157.90/57.19
157.90/57.19

(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.

FIRST(mark(z0), z1) → c8(FIRST(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.90/57.19
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.90/57.19
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.90/57.19
S(mark(z0)) → c11(S(z0)) 157.90/57.19
S(ok(z0)) → c12(S(z0)) 157.90/57.19
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.90/57.19
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.90/57.19
FROM(mark(z0)) → c15(FROM(z0)) 157.90/57.19
FROM(ok(z0)) → c16(FROM(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 157.90/57.19

POL(CONS(x1, x2)) = [3]x1 + [3]x2    157.90/57.19
POL(FIRST(x1, x2)) = [4]x1    157.90/57.19
POL(FROM(x1)) = [5]x1    157.90/57.19
POL(S(x1)) = [3]x1    157.90/57.19
POL(c10(x1)) = x1    157.90/57.19
POL(c11(x1)) = x1    157.90/57.19
POL(c12(x1)) = x1    157.90/57.19
POL(c13(x1)) = x1    157.90/57.19
POL(c14(x1)) = x1    157.90/57.19
POL(c15(x1)) = x1    157.90/57.19
POL(c16(x1)) = x1    157.90/57.19
POL(c8(x1)) = x1    157.90/57.19
POL(c9(x1)) = x1    157.90/57.19
POL(mark(x1)) = [1] + x1    157.90/57.19
POL(ok(x1)) = x1   
157.90/57.19
157.90/57.19

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil) 157.90/57.19
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 157.90/57.19
active(from(z0)) → mark(cons(z0, from(s(z0)))) 157.90/57.19
active(first(z0, z1)) → first(active(z0), z1) 157.90/57.19
active(first(z0, z1)) → first(z0, active(z1)) 157.90/57.19
active(s(z0)) → s(active(z0)) 157.90/57.19
active(cons(z0, z1)) → cons(active(z0), z1) 157.90/57.19
active(from(z0)) → from(active(z0)) 157.90/57.19
first(mark(z0), z1) → mark(first(z0, z1)) 157.90/57.19
first(z0, mark(z1)) → mark(first(z0, z1)) 157.90/57.19
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 157.90/57.19
s(mark(z0)) → mark(s(z0)) 157.90/57.19
s(ok(z0)) → ok(s(z0)) 157.90/57.19
cons(mark(z0), z1) → mark(cons(z0, z1)) 157.90/57.19
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 157.90/57.19
from(mark(z0)) → mark(from(z0)) 157.90/57.19
from(ok(z0)) → ok(from(z0)) 157.90/57.19
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 157.90/57.19
proper(0) → ok(0) 157.90/57.19
proper(nil) → ok(nil) 157.90/57.19
proper(s(z0)) → s(proper(z0)) 157.90/57.19
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 157.90/57.19
proper(from(z0)) → from(proper(z0)) 157.90/57.19
top(mark(z0)) → top(proper(z0)) 157.90/57.19
top(ok(z0)) → top(active(z0))
Tuples:

FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.90/57.19
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.90/57.19
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.90/57.19
S(mark(z0)) → c11(S(z0)) 157.90/57.19
S(ok(z0)) → c12(S(z0)) 157.90/57.19
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.90/57.19
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.90/57.19
FROM(mark(z0)) → c15(FROM(z0)) 157.90/57.19
FROM(ok(z0)) → c16(FROM(z0))
S tuples:

FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.90/57.19
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1))
K tuples:

S(mark(z0)) → c11(S(z0)) 157.90/57.19
S(ok(z0)) → c12(S(z0)) 157.90/57.19
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.90/57.19
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.90/57.19
FROM(mark(z0)) → c15(FROM(z0)) 157.90/57.19
FROM(ok(z0)) → c16(FROM(z0)) 157.90/57.19
FIRST(mark(z0), z1) → c8(FIRST(z0, z1))
Defined Rule Symbols:

active, first, s, cons, from, proper, top

Defined Pair Symbols:

FIRST, S, CONS, FROM

Compound Symbols:

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

157.90/57.19
157.90/57.19

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

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

FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.90/57.19
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.90/57.19
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.90/57.19
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.90/57.19
S(mark(z0)) → c11(S(z0)) 157.90/57.19
S(ok(z0)) → c12(S(z0)) 157.90/57.19
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.90/57.19
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.90/57.19
FROM(mark(z0)) → c15(FROM(z0)) 157.90/57.19
FROM(ok(z0)) → c16(FROM(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 157.90/57.19

POL(CONS(x1, x2)) = x1 + x2 + x22 + x1·x2 + x12 + x13 + x12·x2 + x1·x22 + x23    157.90/57.19
POL(FIRST(x1, x2)) = x1 + x2 + x1·x2 + x12 + x13 + x1·x22    157.90/57.19
POL(FROM(x1)) = x1 + x12 + x13    157.90/57.19
POL(S(x1)) = x1 + x12 + x13    157.90/57.19
POL(c10(x1)) = x1    157.90/57.19
POL(c11(x1)) = x1    157.90/57.19
POL(c12(x1)) = x1    157.90/57.19
POL(c13(x1)) = x1    157.90/57.19
POL(c14(x1)) = x1    157.90/57.19
POL(c15(x1)) = x1    157.90/57.19
POL(c16(x1)) = x1    157.90/57.19
POL(c8(x1)) = x1    157.90/57.19
POL(c9(x1)) = x1    157.90/57.19
POL(mark(x1)) = [1] + x1    157.90/57.19
POL(ok(x1)) = [1] + x1   
157.90/57.19
157.90/57.19

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil) 157.90/57.19
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2))) 157.90/57.19
active(from(z0)) → mark(cons(z0, from(s(z0)))) 157.90/57.19
active(first(z0, z1)) → first(active(z0), z1) 157.90/57.19
active(first(z0, z1)) → first(z0, active(z1)) 157.90/57.19
active(s(z0)) → s(active(z0)) 157.90/57.19
active(cons(z0, z1)) → cons(active(z0), z1) 157.90/57.19
active(from(z0)) → from(active(z0)) 157.90/57.19
first(mark(z0), z1) → mark(first(z0, z1)) 157.90/57.19
first(z0, mark(z1)) → mark(first(z0, z1)) 157.90/57.19
first(ok(z0), ok(z1)) → ok(first(z0, z1)) 157.90/57.19
s(mark(z0)) → mark(s(z0)) 157.90/57.19
s(ok(z0)) → ok(s(z0)) 157.90/57.19
cons(mark(z0), z1) → mark(cons(z0, z1)) 157.90/57.19
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 157.90/57.19
from(mark(z0)) → mark(from(z0)) 157.90/57.19
from(ok(z0)) → ok(from(z0)) 157.90/57.19
proper(first(z0, z1)) → first(proper(z0), proper(z1)) 157.90/57.19
proper(0) → ok(0) 157.90/57.19
proper(nil) → ok(nil) 157.90/57.19
proper(s(z0)) → s(proper(z0)) 157.90/57.19
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 157.90/57.19
proper(from(z0)) → from(proper(z0)) 157.90/57.19
top(mark(z0)) → top(proper(z0)) 157.90/57.19
top(ok(z0)) → top(active(z0))
Tuples:

FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.90/57.19
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.90/57.19
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1)) 157.90/57.19
S(mark(z0)) → c11(S(z0)) 157.90/57.19
S(ok(z0)) → c12(S(z0)) 157.90/57.19
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.90/57.19
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.90/57.19
FROM(mark(z0)) → c15(FROM(z0)) 157.90/57.19
FROM(ok(z0)) → c16(FROM(z0))
S tuples:none
K tuples:

S(mark(z0)) → c11(S(z0)) 157.90/57.19
S(ok(z0)) → c12(S(z0)) 157.90/57.19
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 157.90/57.19
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 157.90/57.19
FROM(mark(z0)) → c15(FROM(z0)) 157.90/57.19
FROM(ok(z0)) → c16(FROM(z0)) 157.90/57.19
FIRST(mark(z0), z1) → c8(FIRST(z0, z1)) 157.90/57.19
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1)) 157.90/57.19
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1))
Defined Rule Symbols:

active, first, s, cons, from, proper, top

Defined Pair Symbols:

FIRST, S, CONS, FROM

Compound Symbols:

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

157.90/57.19
157.90/57.19

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

The set S is empty
157.90/57.19
157.90/57.19

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

157.90/57.19
157.90/57.19
158.28/57.26 EOF