Runtime Complexity (innermost) proof of /tmp/tmpGa3hVK/Ex6_Luc98

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxTRS

↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)

↳2 CdtProblem

↳3 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

↳4 CdtProblem

↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳6 CdtProblem

↳7 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID), 0 ms)

↳8 CdtProblem

↳9 CdtUsableRulesProof (⇔, 18 ms)

↳10 CdtProblem

↳11 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 741 ms)

↳12 CdtProblem

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

↳14 CdtProblem

↳15 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳16 CdtProblem

↳17 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳18 CdtProblem

↳19 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳20 CdtProblem

↳21 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳22 CdtProblem

↳23 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳24 CdtProblem

↳25 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳26 CdtProblem

↳27 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳28 CdtProblem

↳29 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳30 CdtProblem

↳31 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳32 CdtProblem

↳33 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳34 CdtProblem

↳35 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳36 CdtProblem

↳37 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳38 CdtProblem

↳39 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 17 ms)

↳40 CdtProblem

↳41 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

↳42 CdtProblem

↳43 CdtUnreachableProof (⇔, 0 ms)

↳44 CdtProblem

↳45 CdtUsableRulesProof (⇔, 0 ms)

↳46 CdtProblem

↳47 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 0 ms)

↳48 CdtProblem

↳49 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 0 ms)

↳50 CdtProblem

↳51 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)

↳52 BOUNDS(O(1), O(1))

(0) Obligation:

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

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

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(from(z0)) → from(active(z0))
first(mark(z0), z1) → mark(first(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(first(0, z0)) → c
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(from(z0)) → c2(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(first(z0, z1)) → c3(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c4(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c7(FROM(active(z0)), ACTIVE(z0))
FIRST(mark(z0), z1) → c8(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
FROM(mark(z0)) → c15(FROM(z0))
FROM(ok(z0)) → c16(FROM(z0))
PROPER(first(z0, z1)) → c17(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(0) → c18
PROPER(nil) → c19
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c23(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c24(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(first(0, z0)) → c
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(from(z0)) → c2(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(first(z0, z1)) → c3(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c4(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(from(z0)) → c7(FROM(active(z0)), ACTIVE(z0))
FIRST(mark(z0), z1) → c8(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
FROM(mark(z0)) → c15(FROM(z0))
FROM(ok(z0)) → c16(FROM(z0))
PROPER(first(z0, z1)) → c17(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(0) → c18
PROPER(nil) → c19
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c23(TOP(proper(z0)), PROPER(z0))
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:

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

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

Removed 3 trailing nodes:

ACTIVE(first(0, z0)) → c
PROPER(0) → c18
PROPER(nil) → c19

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(from(z0)) → from(active(z0))
first(mark(z0), z1) → mark(first(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

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

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

Removed 2 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(from(z0)) → from(active(z0))
first(mark(z0), z1) → mark(first(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

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

S tuples:

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

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

Split RHS of tuples not part of any SCC

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(from(z0)) → from(active(z0))
first(mark(z0), z1) → mark(first(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

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

S tuples:

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

(9) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(from(z0)) → from(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))

Tuples:

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

S tuples:

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

K tuples:none
Defined Rule Symbols:

active, cons, first, s, from, proper

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

(11) CdtRuleRemovalProof (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:

active(cons(z0, z1)) → cons(active(z0), z1)
s(mark(z0)) → mark(s(z0))
from(mark(z0)) → mark(from(z0))
first(mark(z0), z1) → mark(first(z0, z1))
active(s(z0)) → s(active(z0))
active(from(z0)) → from(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
active(first(z0, z1)) → first(active(z0), z1)
active(first(0, z0)) → mark(nil)
active(from(z0)) → mark(cons(z0, from(s(z0))))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
first(z0, mark(z1)) → mark(first(z0, z1))
from(ok(z0)) → ok(from(z0))
active(first(z0, z1)) → first(z0, active(z1))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
cons(mark(z0), z1) → mark(cons(z0, z1))
s(ok(z0)) → ok(s(z0))

And the Tuples:

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

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

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

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(from(z0)) → from(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))

Tuples:

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

S tuples:

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

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

(13) 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)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c5(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(first(z0, z1))) → c5(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c5(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(from(z0)) → from(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))

Tuples:

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

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

(15) 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)))
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))))
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0)))

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(from(z0)) → from(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))

Tuples:

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

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

(17) 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)))
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c7(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c7(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(first(z0, z1))) → c7(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(from(first(z0, z1))) → c7(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(from(s(z0))) → c7(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(cons(z0, z1))) → c7(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(from(z0))) → c7(FROM(from(active(z0))), ACTIVE(from(z0)))

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(from(z0)) → from(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))

Tuples:

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

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

(19) 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)))
PROPER(first(x0, 0)) → c17(FIRST(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(first(x0, nil)) → c17(FIRST(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(first(x0, s(z0))) → c17(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(first(x0, cons(z0, z1))) → c17(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(first(x0, from(z0))) → c17(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(first(first(z0, z1), x1)) → c17(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(first(0, x1)) → c17(FIRST(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(first(nil, x1)) → c17(FIRST(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(first(s(z0), x1)) → c17(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(first(cons(z0, z1), x1)) → c17(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(first(from(z0), x1)) → c17(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(from(z0)) → from(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))

Tuples:

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

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

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

Removed 4 trailing tuple parts

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(from(z0)) → from(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))

Tuples:

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

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

(23) 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)))
PROPER(s(0)) → c20(S(ok(0)), PROPER(0))
PROPER(s(nil)) → c20(S(ok(nil)), PROPER(nil))
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(cons(z0, z1))) → c20(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(from(z0))) → c20(S(from(proper(z0))), PROPER(from(z0)))

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(from(z0)) → from(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))

Tuples:

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

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

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

Removed 2 trailing tuple parts

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(from(z0)) → from(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))

Tuples:

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

S tuples:

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

K tuples:

TOP(mark(z0)) → c23(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, cons, first, s, from, proper

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

(27) 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)))
PROPER(cons(x0, 0)) → c21(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(cons(x0, nil)) → c21(CONS(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(cons(x0, s(z0))) → c21(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, cons(z0, z1))) → c21(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, from(z0))) → c21(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(first(z0, z1), x1)) → c21(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(0, x1)) → c21(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(cons(nil, x1)) → c21(CONS(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(cons(s(z0), x1)) → c21(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c21(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c21(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(from(z0)) → from(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))

Tuples:

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

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

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

Removed 4 trailing tuple parts

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(from(z0)) → from(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))

Tuples:

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

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

(31) 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)))
PROPER(from(0)) → c22(FROM(ok(0)), PROPER(0))
PROPER(from(nil)) → c22(FROM(ok(nil)), PROPER(nil))
PROPER(from(s(z0))) → c22(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(cons(z0, z1))) → c22(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(from(z0))) → c22(FROM(from(proper(z0))), PROPER(from(z0)))

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(from(z0)) → from(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))

Tuples:

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

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

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

Removed 2 trailing tuple parts

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(from(z0)) → from(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))

Tuples:

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

S tuples:

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

K tuples:

TOP(mark(z0)) → c23(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, cons, first, s, from, proper

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

(35) 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)))
TOP(mark(0)) → c23(TOP(ok(0)), PROPER(0))
TOP(mark(nil)) → c23(TOP(ok(nil)), PROPER(nil))
TOP(mark(s(z0))) → c23(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(cons(z0, z1))) → c23(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(from(z0))) → c23(TOP(from(proper(z0))), PROPER(from(z0)))

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(from(z0)) → from(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))

Tuples:

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

K tuples:

TOP(mark(z0)) → c23(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, cons, first, s, from, proper

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

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

Removed 2 trailing tuple parts

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(from(z0)) → from(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))

Tuples:

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

S tuples:

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

K tuples:

TOP(mark(z0)) → c23(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, cons, first, s, from, proper

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

(39) 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)))
TOP(ok(first(s(z0), cons(z1, z2)))) → c24(TOP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
TOP(ok(from(z0))) → c24(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(first(z0, z1))) → c24(TOP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
TOP(ok(first(z0, z1))) → c24(TOP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
TOP(ok(s(z0))) → c24(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(cons(z0, z1))) → c24(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(from(z0))) → c24(TOP(from(active(z0))), ACTIVE(from(z0)))

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(from(z0)) → from(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))

Tuples:

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

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

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

Removed 2 trailing nodes:

TOP(mark(nil)) → c23(TOP(ok(nil)))
TOP(mark(0)) → c23(TOP(ok(0)))

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(from(z0)) → from(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))

Tuples:

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

K tuples:none
Defined Rule Symbols:

active, cons, first, s, from, proper

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, c24

(43) 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))
ACTIVE(first(z0, z1)) → c4(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c2(S(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c(FIRST(z0, z2))
ACTIVE(s(first(0, z0))) → c5(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c5(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(first(z0, z1))) → c5(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c5(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(cons(first(0, z0), x1)) → c6(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
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))))
ACTIVE(cons(from(z0), x1)) → c6(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c6(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(from(z0), x1)) → c6(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(from(first(0, z0))) → c7(FROM(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(from(first(s(z0), cons(z1, z2)))) → c7(FROM(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(from(from(z0))) → c7(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(first(z0, z1))) → c7(FROM(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(from(first(z0, z1))) → c7(FROM(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(from(s(z0))) → c7(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(cons(z0, z1))) → c7(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(from(z0))) → c7(FROM(from(active(z0))), ACTIVE(from(z0)))
PROPER(first(x0, first(z0, z1))) → c17(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(first(x0, s(z0))) → c17(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(first(x0, cons(z0, z1))) → c17(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(first(x0, from(z0))) → c17(FIRST(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(first(first(z0, z1), x1)) → c17(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(first(s(z0), x1)) → c17(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(first(cons(z0, z1), x1)) → c17(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(first(from(z0), x1)) → c17(FIRST(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(first(x0, 0)) → c17(FIRST(proper(x0), ok(0)), PROPER(x0))
PROPER(first(x0, nil)) → c17(FIRST(proper(x0), ok(nil)), PROPER(x0))
PROPER(first(0, x1)) → c17(FIRST(ok(0), proper(x1)), PROPER(x1))
PROPER(first(nil, x1)) → c17(FIRST(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(first(z0, z1))) → c20(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(cons(z0, z1))) → c20(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(from(z0))) → c20(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(0)) → c20(S(ok(0)))
PROPER(s(nil)) → c20(S(ok(nil)))
PROPER(cons(x0, first(z0, z1))) → c21(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, s(z0))) → c21(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, cons(z0, z1))) → c21(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, from(z0))) → c21(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(first(z0, z1), x1)) → c21(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c21(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c21(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(from(z0), x1)) → c21(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c21(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c21(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c21(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c21(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(from(first(z0, z1))) → c22(FROM(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(from(s(z0))) → c22(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(cons(z0, z1))) → c22(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(from(z0))) → c22(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(0)) → c22(FROM(ok(0)))
PROPER(from(nil)) → c22(FROM(ok(nil)))
TOP(mark(first(z0, z1))) → c23(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
TOP(mark(s(z0))) → c23(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(cons(z0, z1))) → c23(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(from(z0))) → c23(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(ok(first(0, z0))) → c24(TOP(mark(nil)), ACTIVE(first(0, z0)))
TOP(ok(first(s(z0), cons(z1, z2)))) → c24(TOP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
TOP(ok(from(z0))) → c24(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(first(z0, z1))) → c24(TOP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
TOP(ok(first(z0, z1))) → c24(TOP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
TOP(ok(s(z0))) → c24(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(cons(z0, z1))) → c24(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(from(z0))) → c24(TOP(from(active(z0))), ACTIVE(from(z0)))

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(from(z0)) → from(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))

Tuples:

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

S tuples:

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

K tuples:none
Defined Rule Symbols:

active, cons, first, s, from, proper

Defined Pair Symbols:

FIRST, S, CONS, FROM

Compound Symbols:

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

(45) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(from(z0)) → mark(cons(z0, from(s(z0))))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(s(z0)) → s(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(from(z0)) → from(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(s(z0)) → s(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(from(z0)) → from(proper(z0))

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

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

S tuples:

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

K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

FIRST, S, CONS, FROM

Compound Symbols:

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

(47) CdtRuleRemovalProof (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))
CONS(mark(z0), z1) → c13(CONS(z0, z1))

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

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

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

POL(CONS(x₁, x₂)) = [4]x₁
POL(FIRST(x₁, x₂)) = 0
POL(FROM(x₁)) = 0
POL(S(x₁)) = [5]x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = [4] + x₁
POL(ok(x₁)) = x₁

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

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

S tuples:

FIRST(mark(z0), z1) → c8(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1))
S(ok(z0)) → c12(S(z0))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
FROM(mark(z0)) → c15(FROM(z0))
FROM(ok(z0)) → c16(FROM(z0))

K tuples:

S(mark(z0)) → c11(S(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))

Defined Rule Symbols:none

Defined Pair Symbols:

FIRST, S, CONS, FROM

Compound Symbols:

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

(49) CdtRuleRemovalProof (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))
FIRST(z0, mark(z1)) → c9(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c10(FIRST(z0, z1))
S(ok(z0)) → c12(S(z0))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
FROM(mark(z0)) → c15(FROM(z0))
FROM(ok(z0)) → c16(FROM(z0))

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

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

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

POL(CONS(x₁, x₂)) = [5]x₁ + [4]x₂
POL(FIRST(x₁, x₂)) = [2]x₁ + x₂
POL(FROM(x₁)) = [5]x₁
POL(S(x₁)) = [5]x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = [5] + x₁
POL(ok(x₁)) = [5] + x₁

(50) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

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

S tuples:none
K tuples:

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

Defined Rule Symbols:none

Defined Pair Symbols:

FIRST, S, CONS, FROM

Compound Symbols:

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

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

The set S is empty

(0) Obligation:

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

(2) Obligation:

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

(4) Obligation:

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

(6) Obligation:

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

(8) Obligation:

(9) CdtUsableRulesProof (EQUIVALENT transformation)

(10) Obligation:

(11) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

(12) Obligation:

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

(14) Obligation:

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

(16) Obligation:

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

(18) Obligation:

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

(20) Obligation:

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

(22) Obligation:

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

(24) Obligation:

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

(26) Obligation:

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

(28) Obligation:

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

(30) Obligation:

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

(32) Obligation:

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

(34) Obligation:

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

(36) Obligation:

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

(38) Obligation:

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

(40) Obligation:

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

(42) Obligation:

(43) CdtUnreachableProof (EQUIVALENT transformation)

(44) Obligation:

(45) CdtUsableRulesProof (EQUIVALENT transformation)

(46) Obligation:

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

(48) Obligation:

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

(50) Obligation:

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

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