Runtime Complexity (innermost) proof of /tmp/tmpoQex7V/Ex14_AEGL02

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

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 (⇔, 16 ms)

↳10 CdtProblem

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

↳12 CdtProblem

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

↳14 CdtProblem

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

↳16 CdtProblem

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

↳18 CdtProblem

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

↳20 CdtProblem

↳21 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID), 2 ms)

↳22 CdtProblem

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

↳24 CdtProblem

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

↳26 CdtProblem

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

↳28 CdtProblem

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

↳30 CdtProblem

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

↳32 CdtProblem

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

↳34 CdtProblem

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

↳36 CdtProblem

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

↳38 CdtProblem

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

↳40 CdtProblem

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

↳42 CdtProblem

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

↳44 CdtProblem

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

↳46 CdtProblem

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

↳48 CdtProblem

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

↳50 CdtProblem

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

↳52 CdtProblem

↳53 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 670 ms)

↳54 CdtProblem

↳55 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 1483 ms)

↳56 CdtProblem

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

↳58 CdtProblem

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

↳60 CdtProblem

↳61 CdtUnreachableProof (⇔, 0 ms)

↳62 CdtProblem

↳63 CdtUsableRulesProof (⇔, 0 ms)

↳64 CdtProblem

↳65 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 6 ms)

↳66 CdtProblem

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

↳68 CdtProblem

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

↳70 CdtProblem

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

↳72 CdtProblem

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

↳74 CdtProblem

↳75 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^2))), 0 ms)

↳76 CdtProblem

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

↳78 CdtProblem

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

↳80 CdtProblem

↳81 SIsEmptyProof (BOTH BOUNDS(ID, ID), 1 ms)

↳82 BOUNDS(O(1), O(1))

(0) Obligation:

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

active(from(X)) → mark(cons(X, from(s(X))))
active(length(nil)) → mark(0)
active(length(cons(X, Y))) → mark(s(length1(Y)))
active(length1(X)) → mark(length(X))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
proper(0) → ok(0)
proper(length1(X)) → length1(proper(X))
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
length1(ok(X)) → ok(length1(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(from(z0)) → mark(cons(z0, from(s(z0))))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length1(z1)))
active(length1(z0)) → mark(length(z0))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(0) → ok(0)
proper(length1(z0)) → length1(proper(z0))
length(ok(z0)) → ok(length(z0))
length1(ok(z0)) → ok(length1(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(from(z0)) → c(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(length(nil)) → c1
ACTIVE(length(cons(z0, z1))) → c2(S(length1(z1)), LENGTH1(z1))
ACTIVE(length1(z0)) → c3(LENGTH(z0))
ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c5(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(from(z0)) → c13(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c14(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c15(S(proper(z0)), PROPER(z0))
PROPER(length(z0)) → c16(LENGTH(proper(z0)), PROPER(z0))
PROPER(nil) → c17
PROPER(0) → c18
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(from(z0)) → c(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(length(nil)) → c1
ACTIVE(length(cons(z0, z1))) → c2(S(length1(z1)), LENGTH1(z1))
ACTIVE(length1(z0)) → c3(LENGTH(z0))
ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c5(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(from(z0)) → c13(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c14(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c15(S(proper(z0)), PROPER(z0))
PROPER(length(z0)) → c16(LENGTH(proper(z0)), PROPER(z0))
PROPER(nil) → c17
PROPER(0) → c18
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, from, cons, s, proper, length, length1, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, PROPER, LENGTH, LENGTH1, 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

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

Removed 3 trailing nodes:

PROPER(nil) → c17
ACTIVE(length(nil)) → c1
PROPER(0) → c18

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length1(z1)))
active(length1(z0)) → mark(length(z0))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(0) → ok(0)
proper(length1(z0)) → length1(proper(z0))
length(ok(z0)) → ok(length(z0))
length1(ok(z0)) → ok(length1(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(from(z0)) → c(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(length(cons(z0, z1))) → c2(S(length1(z1)), LENGTH1(z1))
ACTIVE(length1(z0)) → c3(LENGTH(z0))
ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c5(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(from(z0)) → c13(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c14(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c15(S(proper(z0)), PROPER(z0))
PROPER(length(z0)) → c16(LENGTH(proper(z0)), PROPER(z0))
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(from(z0)) → c(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(length(cons(z0, z1))) → c2(S(length1(z1)), LENGTH1(z1))
ACTIVE(length1(z0)) → c3(LENGTH(z0))
ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c5(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(from(z0)) → c13(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c14(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c15(S(proper(z0)), PROPER(z0))
PROPER(length(z0)) → c16(LENGTH(proper(z0)), PROPER(z0))
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, from, cons, s, proper, length, length1, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, PROPER, LENGTH, LENGTH1, TOP

Compound Symbols:

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

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

Removed 2 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length1(z1)))
active(length1(z0)) → mark(length(z0))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(0) → ok(0)
proper(length1(z0)) → length1(proper(z0))
length(ok(z0)) → ok(length(z0))
length1(ok(z0)) → ok(length1(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(length(cons(z0, z1))) → c2(S(length1(z1)), LENGTH1(z1))
ACTIVE(length1(z0)) → c3(LENGTH(z0))
ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c5(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(from(z0)) → c13(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c14(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c15(S(proper(z0)), PROPER(z0))
PROPER(length(z0)) → c16(LENGTH(proper(z0)), PROPER(z0))
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))

S tuples:

ACTIVE(length(cons(z0, z1))) → c2(S(length1(z1)), LENGTH1(z1))
ACTIVE(length1(z0)) → c3(LENGTH(z0))
ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c5(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(from(z0)) → c13(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c14(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c15(S(proper(z0)), PROPER(z0))
PROPER(length(z0)) → c16(LENGTH(proper(z0)), PROPER(z0))
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))

K tuples:none
Defined Rule Symbols:

active, from, cons, s, proper, length, length1, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, PROPER, LENGTH, LENGTH1, TOP

Compound Symbols:

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

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

Split RHS of tuples not part of any SCC

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length1(z1)))
active(length1(z0)) → mark(length(z0))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(0) → ok(0)
proper(length1(z0)) → length1(proper(z0))
length(ok(z0)) → ok(length(z0))
length1(ok(z0)) → ok(length1(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c5(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(from(z0)) → c13(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c14(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c15(S(proper(z0)), PROPER(z0))
PROPER(length(z0)) → c16(LENGTH(proper(z0)), PROPER(z0))
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))

S tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c5(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(from(z0)) → c13(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c14(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c15(S(proper(z0)), PROPER(z0))
PROPER(length(z0)) → c16(LENGTH(proper(z0)), PROPER(z0))
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))

K tuples:none
Defined Rule Symbols:

active, from, cons, s, proper, length, length1, top

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, PROPER, LENGTH, LENGTH1, TOP

Compound Symbols:

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

(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(from(z0)) → mark(cons(z0, from(s(z0))))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length1(z1)))
active(length1(z0)) → mark(length(z0))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
length1(ok(z0)) → ok(length1(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(0) → ok(0)
proper(length1(z0)) → length1(proper(z0))
length(ok(z0)) → ok(length(z0))

Tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c5(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(from(z0)) → c13(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c14(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c15(S(proper(z0)), PROPER(z0))
PROPER(length(z0)) → c16(LENGTH(proper(z0)), PROPER(z0))
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))

S tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c5(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(from(z0)) → c13(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c14(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c15(S(proper(z0)), PROPER(z0))
PROPER(length(z0)) → c16(LENGTH(proper(z0)), PROPER(z0))
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))

K tuples:none
Defined Rule Symbols:

active, s, length1, from, cons, proper, length

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, PROPER, LENGTH, LENGTH1, TOP

Compound Symbols:

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

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

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

ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)), ACTIVE(length(nil)))
ACTIVE(from(length(cons(z0, z1)))) → c4(FROM(mark(s(length1(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c4(FROM(mark(length(z0))), ACTIVE(length1(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length1(z1)))
active(length1(z0)) → mark(length(z0))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
length1(ok(z0)) → ok(length1(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(0) → ok(0)
proper(length1(z0)) → length1(proper(z0))
length(ok(z0)) → ok(length(z0))

Tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
ACTIVE(cons(z0, z1)) → c5(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(from(z0)) → c13(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c14(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c15(S(proper(z0)), PROPER(z0))
PROPER(length(z0)) → c16(LENGTH(proper(z0)), PROPER(z0))
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)), ACTIVE(length(nil)))
ACTIVE(from(length(cons(z0, z1)))) → c4(FROM(mark(s(length1(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c4(FROM(mark(length(z0))), ACTIVE(length1(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))

S tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
ACTIVE(cons(z0, z1)) → c5(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(from(z0)) → c13(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c14(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c15(S(proper(z0)), PROPER(z0))
PROPER(length(z0)) → c16(LENGTH(proper(z0)), PROPER(z0))
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)), ACTIVE(length(nil)))
ACTIVE(from(length(cons(z0, z1)))) → c4(FROM(mark(s(length1(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c4(FROM(mark(length(z0))), ACTIVE(length1(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))

K tuples:none
Defined Rule Symbols:

active, s, length1, from, cons, proper, length

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, PROPER, LENGTH, LENGTH1, TOP

Compound Symbols:

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

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

Removed 1 trailing tuple parts

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length1(z1)))
active(length1(z0)) → mark(length(z0))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
length1(ok(z0)) → ok(length1(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(0) → ok(0)
proper(length1(z0)) → length1(proper(z0))
length(ok(z0)) → ok(length(z0))

Tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
ACTIVE(cons(z0, z1)) → c5(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(from(z0)) → c13(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c14(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c15(S(proper(z0)), PROPER(z0))
PROPER(length(z0)) → c16(LENGTH(proper(z0)), PROPER(z0))
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(length(cons(z0, z1)))) → c4(FROM(mark(s(length1(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c4(FROM(mark(length(z0))), ACTIVE(length1(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))

S tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
ACTIVE(cons(z0, z1)) → c5(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(from(z0)) → c13(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c14(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c15(S(proper(z0)), PROPER(z0))
PROPER(length(z0)) → c16(LENGTH(proper(z0)), PROPER(z0))
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(length(cons(z0, z1)))) → c4(FROM(mark(s(length1(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c4(FROM(mark(length(z0))), ACTIVE(length1(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))

K tuples:none
Defined Rule Symbols:

active, s, length1, from, cons, proper, length

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, PROPER, LENGTH, LENGTH1, TOP

Compound Symbols:

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

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

Split RHS of tuples not part of any SCC

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length1(z1)))
active(length1(z0)) → mark(length(z0))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
length1(ok(z0)) → ok(length1(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(0) → ok(0)
proper(length1(z0)) → length1(proper(z0))
length(ok(z0)) → ok(length(z0))

Tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
ACTIVE(cons(z0, z1)) → c5(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(from(z0)) → c13(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c14(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c15(S(proper(z0)), PROPER(z0))
PROPER(length(z0)) → c16(LENGTH(proper(z0)), PROPER(z0))
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))

S tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
ACTIVE(cons(z0, z1)) → c5(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(from(z0)) → c13(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c14(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c15(S(proper(z0)), PROPER(z0))
PROPER(length(z0)) → c16(LENGTH(proper(z0)), PROPER(z0))
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))

K tuples:none
Defined Rule Symbols:

active, s, length1, from, cons, proper, length

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, PROPER, LENGTH, LENGTH1, TOP

Compound Symbols:

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

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

Use narrowing to replace ACTIVE(cons(z0, z1)) → c5(CONS(active(z0), z1), ACTIVE(z0)) by

ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1), ACTIVE(length(nil)))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c5(CONS(mark(s(length1(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c5(CONS(mark(length(z0)), x1), ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length1(z1)))
active(length1(z0)) → mark(length(z0))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
length1(ok(z0)) → ok(length1(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(0) → ok(0)
proper(length1(z0)) → length1(proper(z0))
length(ok(z0)) → ok(length(z0))

Tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(from(z0)) → c13(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c14(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c15(S(proper(z0)), PROPER(z0))
PROPER(length(z0)) → c16(LENGTH(proper(z0)), PROPER(z0))
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1), ACTIVE(length(nil)))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c5(CONS(mark(s(length1(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c5(CONS(mark(length(z0)), x1), ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))

S tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(from(z0)) → c13(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c14(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c15(S(proper(z0)), PROPER(z0))
PROPER(length(z0)) → c16(LENGTH(proper(z0)), PROPER(z0))
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1), ACTIVE(length(nil)))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c5(CONS(mark(s(length1(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c5(CONS(mark(length(z0)), x1), ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))

K tuples:none
Defined Rule Symbols:

active, s, length1, from, cons, proper, length

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, PROPER, LENGTH, LENGTH1, TOP

Compound Symbols:

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

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

Removed 1 trailing tuple parts

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length1(z1)))
active(length1(z0)) → mark(length(z0))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
length1(ok(z0)) → ok(length1(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(0) → ok(0)
proper(length1(z0)) → length1(proper(z0))
length(ok(z0)) → ok(length(z0))

Tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(from(z0)) → c13(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c14(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c15(S(proper(z0)), PROPER(z0))
PROPER(length(z0)) → c16(LENGTH(proper(z0)), PROPER(z0))
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c5(CONS(mark(s(length1(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c5(CONS(mark(length(z0)), x1), ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))

S tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(from(z0)) → c13(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c14(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c15(S(proper(z0)), PROPER(z0))
PROPER(length(z0)) → c16(LENGTH(proper(z0)), PROPER(z0))
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c5(CONS(mark(s(length1(z1))), x1), ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c5(CONS(mark(length(z0)), x1), ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))

K tuples:none
Defined Rule Symbols:

active, s, length1, from, cons, proper, length

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, PROPER, LENGTH, LENGTH1, TOP

Compound Symbols:

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

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

Split RHS of tuples not part of any SCC

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length1(z1)))
active(length1(z0)) → mark(length(z0))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
length1(ok(z0)) → ok(length1(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(0) → ok(0)
proper(length1(z0)) → length1(proper(z0))
length(ok(z0)) → ok(length(z0))

Tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(from(z0)) → c13(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c14(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c15(S(proper(z0)), PROPER(z0))
PROPER(length(z0)) → c16(LENGTH(proper(z0)), PROPER(z0))
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(CONS(mark(s(length1(z1))), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c17(CONS(mark(length(z0)), x1))
ACTIVE(cons(length1(z0), x1)) → c17(ACTIVE(length1(z0)))

S tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(from(z0)) → c13(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c14(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c15(S(proper(z0)), PROPER(z0))
PROPER(length(z0)) → c16(LENGTH(proper(z0)), PROPER(z0))
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(CONS(mark(s(length1(z1))), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c17(CONS(mark(length(z0)), x1))
ACTIVE(cons(length1(z0), x1)) → c17(ACTIVE(length1(z0)))

K tuples:none
Defined Rule Symbols:

active, s, length1, from, cons, proper, length

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, PROPER, LENGTH, LENGTH1, TOP

Compound Symbols:

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

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

Use narrowing to replace ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0)) by

ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(length(nil))) → c6(S(mark(0)), ACTIVE(length(nil)))
ACTIVE(s(length(cons(z0, z1)))) → c6(S(mark(s(length1(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(s(length1(z0))) → c6(S(mark(length(z0))), ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length1(z1)))
active(length1(z0)) → mark(length(z0))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
length1(ok(z0)) → ok(length1(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(0) → ok(0)
proper(length1(z0)) → length1(proper(z0))
length(ok(z0)) → ok(length(z0))

Tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(from(z0)) → c13(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c14(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c15(S(proper(z0)), PROPER(z0))
PROPER(length(z0)) → c16(LENGTH(proper(z0)), PROPER(z0))
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(CONS(mark(s(length1(z1))), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c17(CONS(mark(length(z0)), x1))
ACTIVE(cons(length1(z0), x1)) → c17(ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(length(nil))) → c6(S(mark(0)), ACTIVE(length(nil)))
ACTIVE(s(length(cons(z0, z1)))) → c6(S(mark(s(length1(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(s(length1(z0))) → c6(S(mark(length(z0))), ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))

S tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(from(z0)) → c13(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c14(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c15(S(proper(z0)), PROPER(z0))
PROPER(length(z0)) → c16(LENGTH(proper(z0)), PROPER(z0))
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(CONS(mark(s(length1(z1))), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c17(CONS(mark(length(z0)), x1))
ACTIVE(cons(length1(z0), x1)) → c17(ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(length(nil))) → c6(S(mark(0)), ACTIVE(length(nil)))
ACTIVE(s(length(cons(z0, z1)))) → c6(S(mark(s(length1(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(s(length1(z0))) → c6(S(mark(length(z0))), ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))

K tuples:none
Defined Rule Symbols:

active, s, length1, from, cons, proper, length

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, PROPER, LENGTH, LENGTH1, TOP

Compound Symbols:

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

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

Removed 1 trailing tuple parts

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length1(z1)))
active(length1(z0)) → mark(length(z0))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
length1(ok(z0)) → ok(length1(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(0) → ok(0)
proper(length1(z0)) → length1(proper(z0))
length(ok(z0)) → ok(length(z0))

Tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(from(z0)) → c13(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c14(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c15(S(proper(z0)), PROPER(z0))
PROPER(length(z0)) → c16(LENGTH(proper(z0)), PROPER(z0))
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(CONS(mark(s(length1(z1))), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c17(CONS(mark(length(z0)), x1))
ACTIVE(cons(length1(z0), x1)) → c17(ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(length(cons(z0, z1)))) → c6(S(mark(s(length1(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(s(length1(z0))) → c6(S(mark(length(z0))), ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(length(nil))) → c6(S(mark(0)))

S tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(from(z0)) → c13(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c14(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c15(S(proper(z0)), PROPER(z0))
PROPER(length(z0)) → c16(LENGTH(proper(z0)), PROPER(z0))
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(CONS(mark(s(length1(z1))), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c17(CONS(mark(length(z0)), x1))
ACTIVE(cons(length1(z0), x1)) → c17(ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(length(cons(z0, z1)))) → c6(S(mark(s(length1(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(s(length1(z0))) → c6(S(mark(length(z0))), ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(length(nil))) → c6(S(mark(0)))

K tuples:none
Defined Rule Symbols:

active, s, length1, from, cons, proper, length

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, PROPER, LENGTH, LENGTH1, TOP

Compound Symbols:

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

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

Split RHS of tuples not part of any SCC

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length1(z1)))
active(length1(z0)) → mark(length(z0))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
length1(ok(z0)) → ok(length1(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(0) → ok(0)
proper(length1(z0)) → length1(proper(z0))
length(ok(z0)) → ok(length(z0))

Tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(from(z0)) → c13(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c14(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c15(S(proper(z0)), PROPER(z0))
PROPER(length(z0)) → c16(LENGTH(proper(z0)), PROPER(z0))
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(CONS(mark(s(length1(z1))), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c17(CONS(mark(length(z0)), x1))
ACTIVE(cons(length1(z0), x1)) → c17(ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(length(nil))) → c6(S(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c18(S(mark(s(length1(z1)))))
ACTIVE(s(length(cons(z0, z1)))) → c18(ACTIVE(length(cons(z0, z1))))
ACTIVE(s(length1(z0))) → c18(S(mark(length(z0))))
ACTIVE(s(length1(z0))) → c18(ACTIVE(length1(z0)))

S tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(from(z0)) → c13(FROM(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c14(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c15(S(proper(z0)), PROPER(z0))
PROPER(length(z0)) → c16(LENGTH(proper(z0)), PROPER(z0))
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(CONS(mark(s(length1(z1))), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c17(CONS(mark(length(z0)), x1))
ACTIVE(cons(length1(z0), x1)) → c17(ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(length(nil))) → c6(S(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c18(S(mark(s(length1(z1)))))
ACTIVE(s(length(cons(z0, z1)))) → c18(ACTIVE(length(cons(z0, z1))))
ACTIVE(s(length1(z0))) → c18(S(mark(length(z0))))
ACTIVE(s(length1(z0))) → c18(ACTIVE(length1(z0)))

K tuples:none
Defined Rule Symbols:

active, s, length1, from, cons, proper, length

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, PROPER, LENGTH, LENGTH1, TOP

Compound Symbols:

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

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

Use narrowing to replace PROPER(from(z0)) → c13(FROM(proper(z0)), PROPER(z0)) by

PROPER(from(from(z0))) → c13(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c13(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c13(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(length(z0))) → c13(FROM(length(proper(z0))), PROPER(length(z0)))
PROPER(from(nil)) → c13(FROM(ok(nil)), PROPER(nil))
PROPER(from(0)) → c13(FROM(ok(0)), PROPER(0))
PROPER(from(length1(z0))) → c13(FROM(length1(proper(z0))), PROPER(length1(z0)))

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length1(z1)))
active(length1(z0)) → mark(length(z0))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
length1(ok(z0)) → ok(length1(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(0) → ok(0)
proper(length1(z0)) → length1(proper(z0))
length(ok(z0)) → ok(length(z0))

Tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(cons(z0, z1)) → c14(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c15(S(proper(z0)), PROPER(z0))
PROPER(length(z0)) → c16(LENGTH(proper(z0)), PROPER(z0))
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(CONS(mark(s(length1(z1))), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c17(CONS(mark(length(z0)), x1))
ACTIVE(cons(length1(z0), x1)) → c17(ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(length(nil))) → c6(S(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c18(S(mark(s(length1(z1)))))
ACTIVE(s(length(cons(z0, z1)))) → c18(ACTIVE(length(cons(z0, z1))))
ACTIVE(s(length1(z0))) → c18(S(mark(length(z0))))
ACTIVE(s(length1(z0))) → c18(ACTIVE(length1(z0)))
PROPER(from(from(z0))) → c13(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c13(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c13(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(length(z0))) → c13(FROM(length(proper(z0))), PROPER(length(z0)))
PROPER(from(nil)) → c13(FROM(ok(nil)), PROPER(nil))
PROPER(from(0)) → c13(FROM(ok(0)), PROPER(0))
PROPER(from(length1(z0))) → c13(FROM(length1(proper(z0))), PROPER(length1(z0)))

S tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(cons(z0, z1)) → c14(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c15(S(proper(z0)), PROPER(z0))
PROPER(length(z0)) → c16(LENGTH(proper(z0)), PROPER(z0))
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(CONS(mark(s(length1(z1))), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c17(CONS(mark(length(z0)), x1))
ACTIVE(cons(length1(z0), x1)) → c17(ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(length(nil))) → c6(S(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c18(S(mark(s(length1(z1)))))
ACTIVE(s(length(cons(z0, z1)))) → c18(ACTIVE(length(cons(z0, z1))))
ACTIVE(s(length1(z0))) → c18(S(mark(length(z0))))
ACTIVE(s(length1(z0))) → c18(ACTIVE(length1(z0)))
PROPER(from(from(z0))) → c13(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c13(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c13(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(length(z0))) → c13(FROM(length(proper(z0))), PROPER(length(z0)))
PROPER(from(nil)) → c13(FROM(ok(nil)), PROPER(nil))
PROPER(from(0)) → c13(FROM(ok(0)), PROPER(0))
PROPER(from(length1(z0))) → c13(FROM(length1(proper(z0))), PROPER(length1(z0)))

K tuples:none
Defined Rule Symbols:

active, s, length1, from, cons, proper, length

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, PROPER, LENGTH, LENGTH1, TOP

Compound Symbols:

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

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

Removed 2 trailing tuple parts

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length1(z1)))
active(length1(z0)) → mark(length(z0))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
length1(ok(z0)) → ok(length1(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(0) → ok(0)
proper(length1(z0)) → length1(proper(z0))
length(ok(z0)) → ok(length(z0))

Tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(cons(z0, z1)) → c14(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c15(S(proper(z0)), PROPER(z0))
PROPER(length(z0)) → c16(LENGTH(proper(z0)), PROPER(z0))
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(CONS(mark(s(length1(z1))), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c17(CONS(mark(length(z0)), x1))
ACTIVE(cons(length1(z0), x1)) → c17(ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(length(nil))) → c6(S(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c18(S(mark(s(length1(z1)))))
ACTIVE(s(length(cons(z0, z1)))) → c18(ACTIVE(length(cons(z0, z1))))
ACTIVE(s(length1(z0))) → c18(S(mark(length(z0))))
ACTIVE(s(length1(z0))) → c18(ACTIVE(length1(z0)))
PROPER(from(from(z0))) → c13(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c13(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c13(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(length(z0))) → c13(FROM(length(proper(z0))), PROPER(length(z0)))
PROPER(from(length1(z0))) → c13(FROM(length1(proper(z0))), PROPER(length1(z0)))
PROPER(from(nil)) → c13(FROM(ok(nil)))
PROPER(from(0)) → c13(FROM(ok(0)))

S tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(cons(z0, z1)) → c14(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c15(S(proper(z0)), PROPER(z0))
PROPER(length(z0)) → c16(LENGTH(proper(z0)), PROPER(z0))
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(CONS(mark(s(length1(z1))), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c17(CONS(mark(length(z0)), x1))
ACTIVE(cons(length1(z0), x1)) → c17(ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(length(nil))) → c6(S(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c18(S(mark(s(length1(z1)))))
ACTIVE(s(length(cons(z0, z1)))) → c18(ACTIVE(length(cons(z0, z1))))
ACTIVE(s(length1(z0))) → c18(S(mark(length(z0))))
ACTIVE(s(length1(z0))) → c18(ACTIVE(length1(z0)))
PROPER(from(from(z0))) → c13(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c13(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c13(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(length(z0))) → c13(FROM(length(proper(z0))), PROPER(length(z0)))
PROPER(from(length1(z0))) → c13(FROM(length1(proper(z0))), PROPER(length1(z0)))
PROPER(from(nil)) → c13(FROM(ok(nil)))
PROPER(from(0)) → c13(FROM(ok(0)))

K tuples:none
Defined Rule Symbols:

active, s, length1, from, cons, proper, length

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, PROPER, LENGTH, LENGTH1, TOP

Compound Symbols:

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

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

Use narrowing to replace PROPER(cons(z0, z1)) → c14(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(cons(x0, from(z0))) → c14(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c14(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c14(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, length(z0))) → c14(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, nil)) → c14(CONS(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(cons(x0, 0)) → c14(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(cons(x0, length1(z0))) → c14(CONS(proper(x0), length1(proper(z0))), PROPER(x0), PROPER(length1(z0)))
PROPER(cons(from(z0), x1)) → c14(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c14(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c14(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c14(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(nil, x1)) → c14(CONS(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(cons(0, x1)) → c14(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(cons(length1(z0), x1)) → c14(CONS(length1(proper(z0)), proper(x1)), PROPER(length1(z0)), PROPER(x1))

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length1(z1)))
active(length1(z0)) → mark(length(z0))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
length1(ok(z0)) → ok(length1(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(0) → ok(0)
proper(length1(z0)) → length1(proper(z0))
length(ok(z0)) → ok(length(z0))

Tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(s(z0)) → c15(S(proper(z0)), PROPER(z0))
PROPER(length(z0)) → c16(LENGTH(proper(z0)), PROPER(z0))
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(CONS(mark(s(length1(z1))), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c17(CONS(mark(length(z0)), x1))
ACTIVE(cons(length1(z0), x1)) → c17(ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(length(nil))) → c6(S(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c18(S(mark(s(length1(z1)))))
ACTIVE(s(length(cons(z0, z1)))) → c18(ACTIVE(length(cons(z0, z1))))
ACTIVE(s(length1(z0))) → c18(S(mark(length(z0))))
ACTIVE(s(length1(z0))) → c18(ACTIVE(length1(z0)))
PROPER(from(from(z0))) → c13(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c13(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c13(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(length(z0))) → c13(FROM(length(proper(z0))), PROPER(length(z0)))
PROPER(from(length1(z0))) → c13(FROM(length1(proper(z0))), PROPER(length1(z0)))
PROPER(from(nil)) → c13(FROM(ok(nil)))
PROPER(from(0)) → c13(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c14(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c14(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c14(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, length(z0))) → c14(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, nil)) → c14(CONS(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(cons(x0, 0)) → c14(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(cons(x0, length1(z0))) → c14(CONS(proper(x0), length1(proper(z0))), PROPER(x0), PROPER(length1(z0)))
PROPER(cons(from(z0), x1)) → c14(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c14(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c14(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c14(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(nil, x1)) → c14(CONS(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(cons(0, x1)) → c14(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(cons(length1(z0), x1)) → c14(CONS(length1(proper(z0)), proper(x1)), PROPER(length1(z0)), PROPER(x1))

S tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(s(z0)) → c15(S(proper(z0)), PROPER(z0))
PROPER(length(z0)) → c16(LENGTH(proper(z0)), PROPER(z0))
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(CONS(mark(s(length1(z1))), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c17(CONS(mark(length(z0)), x1))
ACTIVE(cons(length1(z0), x1)) → c17(ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(length(nil))) → c6(S(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c18(S(mark(s(length1(z1)))))
ACTIVE(s(length(cons(z0, z1)))) → c18(ACTIVE(length(cons(z0, z1))))
ACTIVE(s(length1(z0))) → c18(S(mark(length(z0))))
ACTIVE(s(length1(z0))) → c18(ACTIVE(length1(z0)))
PROPER(from(from(z0))) → c13(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c13(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c13(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(length(z0))) → c13(FROM(length(proper(z0))), PROPER(length(z0)))
PROPER(from(length1(z0))) → c13(FROM(length1(proper(z0))), PROPER(length1(z0)))
PROPER(from(nil)) → c13(FROM(ok(nil)))
PROPER(from(0)) → c13(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c14(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c14(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c14(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, length(z0))) → c14(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, nil)) → c14(CONS(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(cons(x0, 0)) → c14(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(cons(x0, length1(z0))) → c14(CONS(proper(x0), length1(proper(z0))), PROPER(x0), PROPER(length1(z0)))
PROPER(cons(from(z0), x1)) → c14(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c14(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c14(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c14(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(nil, x1)) → c14(CONS(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(cons(0, x1)) → c14(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(cons(length1(z0), x1)) → c14(CONS(length1(proper(z0)), proper(x1)), PROPER(length1(z0)), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, s, length1, from, cons, proper, length

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, PROPER, LENGTH, LENGTH1, TOP

Compound Symbols:

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

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

Removed 4 trailing tuple parts

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length1(z1)))
active(length1(z0)) → mark(length(z0))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
length1(ok(z0)) → ok(length1(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(0) → ok(0)
proper(length1(z0)) → length1(proper(z0))
length(ok(z0)) → ok(length(z0))

Tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(s(z0)) → c15(S(proper(z0)), PROPER(z0))
PROPER(length(z0)) → c16(LENGTH(proper(z0)), PROPER(z0))
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(CONS(mark(s(length1(z1))), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c17(CONS(mark(length(z0)), x1))
ACTIVE(cons(length1(z0), x1)) → c17(ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(length(nil))) → c6(S(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c18(S(mark(s(length1(z1)))))
ACTIVE(s(length(cons(z0, z1)))) → c18(ACTIVE(length(cons(z0, z1))))
ACTIVE(s(length1(z0))) → c18(S(mark(length(z0))))
ACTIVE(s(length1(z0))) → c18(ACTIVE(length1(z0)))
PROPER(from(from(z0))) → c13(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c13(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c13(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(length(z0))) → c13(FROM(length(proper(z0))), PROPER(length(z0)))
PROPER(from(length1(z0))) → c13(FROM(length1(proper(z0))), PROPER(length1(z0)))
PROPER(from(nil)) → c13(FROM(ok(nil)))
PROPER(from(0)) → c13(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c14(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c14(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c14(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, length(z0))) → c14(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, length1(z0))) → c14(CONS(proper(x0), length1(proper(z0))), PROPER(x0), PROPER(length1(z0)))
PROPER(cons(from(z0), x1)) → c14(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c14(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c14(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c14(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(length1(z0), x1)) → c14(CONS(length1(proper(z0)), proper(x1)), PROPER(length1(z0)), PROPER(x1))
PROPER(cons(x0, nil)) → c14(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, 0)) → c14(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(nil, x1)) → c14(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c14(CONS(ok(0), proper(x1)), PROPER(x1))

S tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(s(z0)) → c15(S(proper(z0)), PROPER(z0))
PROPER(length(z0)) → c16(LENGTH(proper(z0)), PROPER(z0))
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(CONS(mark(s(length1(z1))), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c17(CONS(mark(length(z0)), x1))
ACTIVE(cons(length1(z0), x1)) → c17(ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(length(nil))) → c6(S(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c18(S(mark(s(length1(z1)))))
ACTIVE(s(length(cons(z0, z1)))) → c18(ACTIVE(length(cons(z0, z1))))
ACTIVE(s(length1(z0))) → c18(S(mark(length(z0))))
ACTIVE(s(length1(z0))) → c18(ACTIVE(length1(z0)))
PROPER(from(from(z0))) → c13(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c13(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c13(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(length(z0))) → c13(FROM(length(proper(z0))), PROPER(length(z0)))
PROPER(from(length1(z0))) → c13(FROM(length1(proper(z0))), PROPER(length1(z0)))
PROPER(from(nil)) → c13(FROM(ok(nil)))
PROPER(from(0)) → c13(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c14(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c14(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c14(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, length(z0))) → c14(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, length1(z0))) → c14(CONS(proper(x0), length1(proper(z0))), PROPER(x0), PROPER(length1(z0)))
PROPER(cons(from(z0), x1)) → c14(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c14(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c14(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c14(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(length1(z0), x1)) → c14(CONS(length1(proper(z0)), proper(x1)), PROPER(length1(z0)), PROPER(x1))
PROPER(cons(x0, nil)) → c14(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, 0)) → c14(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(nil, x1)) → c14(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c14(CONS(ok(0), proper(x1)), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, s, length1, from, cons, proper, length

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, PROPER, LENGTH, LENGTH1, TOP

Compound Symbols:

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

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

Use narrowing to replace PROPER(s(z0)) → c15(S(proper(z0)), PROPER(z0)) by

PROPER(s(from(z0))) → c15(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c15(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c15(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(length(z0))) → c15(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(nil)) → c15(S(ok(nil)), PROPER(nil))
PROPER(s(0)) → c15(S(ok(0)), PROPER(0))
PROPER(s(length1(z0))) → c15(S(length1(proper(z0))), PROPER(length1(z0)))

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length1(z1)))
active(length1(z0)) → mark(length(z0))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
length1(ok(z0)) → ok(length1(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(0) → ok(0)
proper(length1(z0)) → length1(proper(z0))
length(ok(z0)) → ok(length(z0))

Tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(length(z0)) → c16(LENGTH(proper(z0)), PROPER(z0))
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(CONS(mark(s(length1(z1))), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c17(CONS(mark(length(z0)), x1))
ACTIVE(cons(length1(z0), x1)) → c17(ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(length(nil))) → c6(S(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c18(S(mark(s(length1(z1)))))
ACTIVE(s(length(cons(z0, z1)))) → c18(ACTIVE(length(cons(z0, z1))))
ACTIVE(s(length1(z0))) → c18(S(mark(length(z0))))
ACTIVE(s(length1(z0))) → c18(ACTIVE(length1(z0)))
PROPER(from(from(z0))) → c13(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c13(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c13(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(length(z0))) → c13(FROM(length(proper(z0))), PROPER(length(z0)))
PROPER(from(length1(z0))) → c13(FROM(length1(proper(z0))), PROPER(length1(z0)))
PROPER(from(nil)) → c13(FROM(ok(nil)))
PROPER(from(0)) → c13(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c14(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c14(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c14(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, length(z0))) → c14(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, length1(z0))) → c14(CONS(proper(x0), length1(proper(z0))), PROPER(x0), PROPER(length1(z0)))
PROPER(cons(from(z0), x1)) → c14(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c14(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c14(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c14(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(length1(z0), x1)) → c14(CONS(length1(proper(z0)), proper(x1)), PROPER(length1(z0)), PROPER(x1))
PROPER(cons(x0, nil)) → c14(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, 0)) → c14(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(nil, x1)) → c14(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c14(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c15(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c15(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c15(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(length(z0))) → c15(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(nil)) → c15(S(ok(nil)), PROPER(nil))
PROPER(s(0)) → c15(S(ok(0)), PROPER(0))
PROPER(s(length1(z0))) → c15(S(length1(proper(z0))), PROPER(length1(z0)))

S tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(length(z0)) → c16(LENGTH(proper(z0)), PROPER(z0))
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(CONS(mark(s(length1(z1))), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c17(CONS(mark(length(z0)), x1))
ACTIVE(cons(length1(z0), x1)) → c17(ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(length(nil))) → c6(S(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c18(S(mark(s(length1(z1)))))
ACTIVE(s(length(cons(z0, z1)))) → c18(ACTIVE(length(cons(z0, z1))))
ACTIVE(s(length1(z0))) → c18(S(mark(length(z0))))
ACTIVE(s(length1(z0))) → c18(ACTIVE(length1(z0)))
PROPER(from(from(z0))) → c13(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c13(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c13(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(length(z0))) → c13(FROM(length(proper(z0))), PROPER(length(z0)))
PROPER(from(length1(z0))) → c13(FROM(length1(proper(z0))), PROPER(length1(z0)))
PROPER(from(nil)) → c13(FROM(ok(nil)))
PROPER(from(0)) → c13(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c14(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c14(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c14(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, length(z0))) → c14(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, length1(z0))) → c14(CONS(proper(x0), length1(proper(z0))), PROPER(x0), PROPER(length1(z0)))
PROPER(cons(from(z0), x1)) → c14(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c14(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c14(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c14(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(length1(z0), x1)) → c14(CONS(length1(proper(z0)), proper(x1)), PROPER(length1(z0)), PROPER(x1))
PROPER(cons(x0, nil)) → c14(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, 0)) → c14(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(nil, x1)) → c14(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c14(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c15(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c15(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c15(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(length(z0))) → c15(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(nil)) → c15(S(ok(nil)), PROPER(nil))
PROPER(s(0)) → c15(S(ok(0)), PROPER(0))
PROPER(s(length1(z0))) → c15(S(length1(proper(z0))), PROPER(length1(z0)))

K tuples:none
Defined Rule Symbols:

active, s, length1, from, cons, proper, length

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, PROPER, LENGTH, LENGTH1, TOP

Compound Symbols:

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

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

Removed 2 trailing tuple parts

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length1(z1)))
active(length1(z0)) → mark(length(z0))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
length1(ok(z0)) → ok(length1(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(0) → ok(0)
proper(length1(z0)) → length1(proper(z0))
length(ok(z0)) → ok(length(z0))

Tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(length(z0)) → c16(LENGTH(proper(z0)), PROPER(z0))
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(CONS(mark(s(length1(z1))), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c17(CONS(mark(length(z0)), x1))
ACTIVE(cons(length1(z0), x1)) → c17(ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(length(nil))) → c6(S(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c18(S(mark(s(length1(z1)))))
ACTIVE(s(length(cons(z0, z1)))) → c18(ACTIVE(length(cons(z0, z1))))
ACTIVE(s(length1(z0))) → c18(S(mark(length(z0))))
ACTIVE(s(length1(z0))) → c18(ACTIVE(length1(z0)))
PROPER(from(from(z0))) → c13(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c13(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c13(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(length(z0))) → c13(FROM(length(proper(z0))), PROPER(length(z0)))
PROPER(from(length1(z0))) → c13(FROM(length1(proper(z0))), PROPER(length1(z0)))
PROPER(from(nil)) → c13(FROM(ok(nil)))
PROPER(from(0)) → c13(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c14(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c14(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c14(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, length(z0))) → c14(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, length1(z0))) → c14(CONS(proper(x0), length1(proper(z0))), PROPER(x0), PROPER(length1(z0)))
PROPER(cons(from(z0), x1)) → c14(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c14(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c14(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c14(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(length1(z0), x1)) → c14(CONS(length1(proper(z0)), proper(x1)), PROPER(length1(z0)), PROPER(x1))
PROPER(cons(x0, nil)) → c14(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, 0)) → c14(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(nil, x1)) → c14(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c14(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c15(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c15(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c15(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(length(z0))) → c15(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(length1(z0))) → c15(S(length1(proper(z0))), PROPER(length1(z0)))
PROPER(s(nil)) → c15(S(ok(nil)))
PROPER(s(0)) → c15(S(ok(0)))

S tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(length(z0)) → c16(LENGTH(proper(z0)), PROPER(z0))
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(CONS(mark(s(length1(z1))), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c17(CONS(mark(length(z0)), x1))
ACTIVE(cons(length1(z0), x1)) → c17(ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(length(nil))) → c6(S(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c18(S(mark(s(length1(z1)))))
ACTIVE(s(length(cons(z0, z1)))) → c18(ACTIVE(length(cons(z0, z1))))
ACTIVE(s(length1(z0))) → c18(S(mark(length(z0))))
ACTIVE(s(length1(z0))) → c18(ACTIVE(length1(z0)))
PROPER(from(from(z0))) → c13(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c13(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c13(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(length(z0))) → c13(FROM(length(proper(z0))), PROPER(length(z0)))
PROPER(from(length1(z0))) → c13(FROM(length1(proper(z0))), PROPER(length1(z0)))
PROPER(from(nil)) → c13(FROM(ok(nil)))
PROPER(from(0)) → c13(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c14(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c14(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c14(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, length(z0))) → c14(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, length1(z0))) → c14(CONS(proper(x0), length1(proper(z0))), PROPER(x0), PROPER(length1(z0)))
PROPER(cons(from(z0), x1)) → c14(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c14(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c14(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c14(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(length1(z0), x1)) → c14(CONS(length1(proper(z0)), proper(x1)), PROPER(length1(z0)), PROPER(x1))
PROPER(cons(x0, nil)) → c14(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, 0)) → c14(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(nil, x1)) → c14(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c14(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c15(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c15(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c15(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(length(z0))) → c15(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(length1(z0))) → c15(S(length1(proper(z0))), PROPER(length1(z0)))
PROPER(s(nil)) → c15(S(ok(nil)))
PROPER(s(0)) → c15(S(ok(0)))

K tuples:none
Defined Rule Symbols:

active, s, length1, from, cons, proper, length

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, PROPER, LENGTH, LENGTH1, TOP

Compound Symbols:

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

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

Use narrowing to replace PROPER(length(z0)) → c16(LENGTH(proper(z0)), PROPER(z0)) by

PROPER(length(from(z0))) → c16(LENGTH(from(proper(z0))), PROPER(from(z0)))
PROPER(length(cons(z0, z1))) → c16(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(s(z0))) → c16(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(length(z0))) → c16(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(nil)) → c16(LENGTH(ok(nil)), PROPER(nil))
PROPER(length(0)) → c16(LENGTH(ok(0)), PROPER(0))
PROPER(length(length1(z0))) → c16(LENGTH(length1(proper(z0))), PROPER(length1(z0)))

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length1(z1)))
active(length1(z0)) → mark(length(z0))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
length1(ok(z0)) → ok(length1(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(0) → ok(0)
proper(length1(z0)) → length1(proper(z0))
length(ok(z0)) → ok(length(z0))

Tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(CONS(mark(s(length1(z1))), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c17(CONS(mark(length(z0)), x1))
ACTIVE(cons(length1(z0), x1)) → c17(ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(length(nil))) → c6(S(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c18(S(mark(s(length1(z1)))))
ACTIVE(s(length(cons(z0, z1)))) → c18(ACTIVE(length(cons(z0, z1))))
ACTIVE(s(length1(z0))) → c18(S(mark(length(z0))))
ACTIVE(s(length1(z0))) → c18(ACTIVE(length1(z0)))
PROPER(from(from(z0))) → c13(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c13(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c13(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(length(z0))) → c13(FROM(length(proper(z0))), PROPER(length(z0)))
PROPER(from(length1(z0))) → c13(FROM(length1(proper(z0))), PROPER(length1(z0)))
PROPER(from(nil)) → c13(FROM(ok(nil)))
PROPER(from(0)) → c13(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c14(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c14(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c14(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, length(z0))) → c14(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, length1(z0))) → c14(CONS(proper(x0), length1(proper(z0))), PROPER(x0), PROPER(length1(z0)))
PROPER(cons(from(z0), x1)) → c14(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c14(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c14(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c14(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(length1(z0), x1)) → c14(CONS(length1(proper(z0)), proper(x1)), PROPER(length1(z0)), PROPER(x1))
PROPER(cons(x0, nil)) → c14(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, 0)) → c14(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(nil, x1)) → c14(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c14(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c15(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c15(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c15(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(length(z0))) → c15(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(length1(z0))) → c15(S(length1(proper(z0))), PROPER(length1(z0)))
PROPER(s(nil)) → c15(S(ok(nil)))
PROPER(s(0)) → c15(S(ok(0)))
PROPER(length(from(z0))) → c16(LENGTH(from(proper(z0))), PROPER(from(z0)))
PROPER(length(cons(z0, z1))) → c16(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(s(z0))) → c16(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(length(z0))) → c16(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(nil)) → c16(LENGTH(ok(nil)), PROPER(nil))
PROPER(length(0)) → c16(LENGTH(ok(0)), PROPER(0))
PROPER(length(length1(z0))) → c16(LENGTH(length1(proper(z0))), PROPER(length1(z0)))

S tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(CONS(mark(s(length1(z1))), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c17(CONS(mark(length(z0)), x1))
ACTIVE(cons(length1(z0), x1)) → c17(ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(length(nil))) → c6(S(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c18(S(mark(s(length1(z1)))))
ACTIVE(s(length(cons(z0, z1)))) → c18(ACTIVE(length(cons(z0, z1))))
ACTIVE(s(length1(z0))) → c18(S(mark(length(z0))))
ACTIVE(s(length1(z0))) → c18(ACTIVE(length1(z0)))
PROPER(from(from(z0))) → c13(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c13(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c13(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(length(z0))) → c13(FROM(length(proper(z0))), PROPER(length(z0)))
PROPER(from(length1(z0))) → c13(FROM(length1(proper(z0))), PROPER(length1(z0)))
PROPER(from(nil)) → c13(FROM(ok(nil)))
PROPER(from(0)) → c13(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c14(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c14(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c14(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, length(z0))) → c14(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, length1(z0))) → c14(CONS(proper(x0), length1(proper(z0))), PROPER(x0), PROPER(length1(z0)))
PROPER(cons(from(z0), x1)) → c14(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c14(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c14(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c14(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(length1(z0), x1)) → c14(CONS(length1(proper(z0)), proper(x1)), PROPER(length1(z0)), PROPER(x1))
PROPER(cons(x0, nil)) → c14(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, 0)) → c14(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(nil, x1)) → c14(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c14(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c15(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c15(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c15(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(length(z0))) → c15(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(length1(z0))) → c15(S(length1(proper(z0))), PROPER(length1(z0)))
PROPER(s(nil)) → c15(S(ok(nil)))
PROPER(s(0)) → c15(S(ok(0)))
PROPER(length(from(z0))) → c16(LENGTH(from(proper(z0))), PROPER(from(z0)))
PROPER(length(cons(z0, z1))) → c16(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(s(z0))) → c16(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(length(z0))) → c16(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(nil)) → c16(LENGTH(ok(nil)), PROPER(nil))
PROPER(length(0)) → c16(LENGTH(ok(0)), PROPER(0))
PROPER(length(length1(z0))) → c16(LENGTH(length1(proper(z0))), PROPER(length1(z0)))

K tuples:none
Defined Rule Symbols:

active, s, length1, from, cons, proper, length

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, PROPER, LENGTH, LENGTH1, TOP

Compound Symbols:

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

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

Removed 2 trailing tuple parts

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length1(z1)))
active(length1(z0)) → mark(length(z0))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
length1(ok(z0)) → ok(length1(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(0) → ok(0)
proper(length1(z0)) → length1(proper(z0))
length(ok(z0)) → ok(length(z0))

Tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(CONS(mark(s(length1(z1))), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c17(CONS(mark(length(z0)), x1))
ACTIVE(cons(length1(z0), x1)) → c17(ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(length(nil))) → c6(S(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c18(S(mark(s(length1(z1)))))
ACTIVE(s(length(cons(z0, z1)))) → c18(ACTIVE(length(cons(z0, z1))))
ACTIVE(s(length1(z0))) → c18(S(mark(length(z0))))
ACTIVE(s(length1(z0))) → c18(ACTIVE(length1(z0)))
PROPER(from(from(z0))) → c13(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c13(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c13(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(length(z0))) → c13(FROM(length(proper(z0))), PROPER(length(z0)))
PROPER(from(length1(z0))) → c13(FROM(length1(proper(z0))), PROPER(length1(z0)))
PROPER(from(nil)) → c13(FROM(ok(nil)))
PROPER(from(0)) → c13(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c14(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c14(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c14(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, length(z0))) → c14(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, length1(z0))) → c14(CONS(proper(x0), length1(proper(z0))), PROPER(x0), PROPER(length1(z0)))
PROPER(cons(from(z0), x1)) → c14(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c14(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c14(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c14(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(length1(z0), x1)) → c14(CONS(length1(proper(z0)), proper(x1)), PROPER(length1(z0)), PROPER(x1))
PROPER(cons(x0, nil)) → c14(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, 0)) → c14(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(nil, x1)) → c14(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c14(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c15(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c15(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c15(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(length(z0))) → c15(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(length1(z0))) → c15(S(length1(proper(z0))), PROPER(length1(z0)))
PROPER(s(nil)) → c15(S(ok(nil)))
PROPER(s(0)) → c15(S(ok(0)))
PROPER(length(from(z0))) → c16(LENGTH(from(proper(z0))), PROPER(from(z0)))
PROPER(length(cons(z0, z1))) → c16(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(s(z0))) → c16(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(length(z0))) → c16(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(length1(z0))) → c16(LENGTH(length1(proper(z0))), PROPER(length1(z0)))
PROPER(length(nil)) → c16(LENGTH(ok(nil)))
PROPER(length(0)) → c16(LENGTH(ok(0)))

S tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(CONS(mark(s(length1(z1))), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c17(CONS(mark(length(z0)), x1))
ACTIVE(cons(length1(z0), x1)) → c17(ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(length(nil))) → c6(S(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c18(S(mark(s(length1(z1)))))
ACTIVE(s(length(cons(z0, z1)))) → c18(ACTIVE(length(cons(z0, z1))))
ACTIVE(s(length1(z0))) → c18(S(mark(length(z0))))
ACTIVE(s(length1(z0))) → c18(ACTIVE(length1(z0)))
PROPER(from(from(z0))) → c13(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c13(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c13(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(length(z0))) → c13(FROM(length(proper(z0))), PROPER(length(z0)))
PROPER(from(length1(z0))) → c13(FROM(length1(proper(z0))), PROPER(length1(z0)))
PROPER(from(nil)) → c13(FROM(ok(nil)))
PROPER(from(0)) → c13(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c14(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c14(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c14(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, length(z0))) → c14(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, length1(z0))) → c14(CONS(proper(x0), length1(proper(z0))), PROPER(x0), PROPER(length1(z0)))
PROPER(cons(from(z0), x1)) → c14(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c14(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c14(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c14(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(length1(z0), x1)) → c14(CONS(length1(proper(z0)), proper(x1)), PROPER(length1(z0)), PROPER(x1))
PROPER(cons(x0, nil)) → c14(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, 0)) → c14(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(nil, x1)) → c14(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c14(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c15(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c15(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c15(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(length(z0))) → c15(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(length1(z0))) → c15(S(length1(proper(z0))), PROPER(length1(z0)))
PROPER(s(nil)) → c15(S(ok(nil)))
PROPER(s(0)) → c15(S(ok(0)))
PROPER(length(from(z0))) → c16(LENGTH(from(proper(z0))), PROPER(from(z0)))
PROPER(length(cons(z0, z1))) → c16(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(s(z0))) → c16(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(length(z0))) → c16(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(length1(z0))) → c16(LENGTH(length1(proper(z0))), PROPER(length1(z0)))
PROPER(length(nil)) → c16(LENGTH(ok(nil)))
PROPER(length(0)) → c16(LENGTH(ok(0)))

K tuples:none
Defined Rule Symbols:

active, s, length1, from, cons, proper, length

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, PROPER, LENGTH, LENGTH1, TOP

Compound Symbols:

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

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

Use narrowing to replace PROPER(length1(z0)) → c19(LENGTH1(proper(z0)), PROPER(z0)) by

PROPER(length1(from(z0))) → c19(LENGTH1(from(proper(z0))), PROPER(from(z0)))
PROPER(length1(cons(z0, z1))) → c19(LENGTH1(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length1(s(z0))) → c19(LENGTH1(s(proper(z0))), PROPER(s(z0)))
PROPER(length1(length(z0))) → c19(LENGTH1(length(proper(z0))), PROPER(length(z0)))
PROPER(length1(nil)) → c19(LENGTH1(ok(nil)), PROPER(nil))
PROPER(length1(0)) → c19(LENGTH1(ok(0)), PROPER(0))
PROPER(length1(length1(z0))) → c19(LENGTH1(length1(proper(z0))), PROPER(length1(z0)))

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length1(z1)))
active(length1(z0)) → mark(length(z0))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
length1(ok(z0)) → ok(length1(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(0) → ok(0)
proper(length1(z0)) → length1(proper(z0))
length(ok(z0)) → ok(length(z0))

Tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(CONS(mark(s(length1(z1))), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c17(CONS(mark(length(z0)), x1))
ACTIVE(cons(length1(z0), x1)) → c17(ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(length(nil))) → c6(S(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c18(S(mark(s(length1(z1)))))
ACTIVE(s(length(cons(z0, z1)))) → c18(ACTIVE(length(cons(z0, z1))))
ACTIVE(s(length1(z0))) → c18(S(mark(length(z0))))
ACTIVE(s(length1(z0))) → c18(ACTIVE(length1(z0)))
PROPER(from(from(z0))) → c13(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c13(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c13(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(length(z0))) → c13(FROM(length(proper(z0))), PROPER(length(z0)))
PROPER(from(length1(z0))) → c13(FROM(length1(proper(z0))), PROPER(length1(z0)))
PROPER(from(nil)) → c13(FROM(ok(nil)))
PROPER(from(0)) → c13(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c14(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c14(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c14(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, length(z0))) → c14(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, length1(z0))) → c14(CONS(proper(x0), length1(proper(z0))), PROPER(x0), PROPER(length1(z0)))
PROPER(cons(from(z0), x1)) → c14(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c14(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c14(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c14(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(length1(z0), x1)) → c14(CONS(length1(proper(z0)), proper(x1)), PROPER(length1(z0)), PROPER(x1))
PROPER(cons(x0, nil)) → c14(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, 0)) → c14(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(nil, x1)) → c14(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c14(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c15(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c15(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c15(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(length(z0))) → c15(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(length1(z0))) → c15(S(length1(proper(z0))), PROPER(length1(z0)))
PROPER(s(nil)) → c15(S(ok(nil)))
PROPER(s(0)) → c15(S(ok(0)))
PROPER(length(from(z0))) → c16(LENGTH(from(proper(z0))), PROPER(from(z0)))
PROPER(length(cons(z0, z1))) → c16(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(s(z0))) → c16(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(length(z0))) → c16(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(length1(z0))) → c16(LENGTH(length1(proper(z0))), PROPER(length1(z0)))
PROPER(length(nil)) → c16(LENGTH(ok(nil)))
PROPER(length(0)) → c16(LENGTH(ok(0)))
PROPER(length1(from(z0))) → c19(LENGTH1(from(proper(z0))), PROPER(from(z0)))
PROPER(length1(cons(z0, z1))) → c19(LENGTH1(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length1(s(z0))) → c19(LENGTH1(s(proper(z0))), PROPER(s(z0)))
PROPER(length1(length(z0))) → c19(LENGTH1(length(proper(z0))), PROPER(length(z0)))
PROPER(length1(nil)) → c19(LENGTH1(ok(nil)), PROPER(nil))
PROPER(length1(0)) → c19(LENGTH1(ok(0)), PROPER(0))
PROPER(length1(length1(z0))) → c19(LENGTH1(length1(proper(z0))), PROPER(length1(z0)))

S tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(CONS(mark(s(length1(z1))), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c17(CONS(mark(length(z0)), x1))
ACTIVE(cons(length1(z0), x1)) → c17(ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(length(nil))) → c6(S(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c18(S(mark(s(length1(z1)))))
ACTIVE(s(length(cons(z0, z1)))) → c18(ACTIVE(length(cons(z0, z1))))
ACTIVE(s(length1(z0))) → c18(S(mark(length(z0))))
ACTIVE(s(length1(z0))) → c18(ACTIVE(length1(z0)))
PROPER(from(from(z0))) → c13(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c13(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c13(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(length(z0))) → c13(FROM(length(proper(z0))), PROPER(length(z0)))
PROPER(from(length1(z0))) → c13(FROM(length1(proper(z0))), PROPER(length1(z0)))
PROPER(from(nil)) → c13(FROM(ok(nil)))
PROPER(from(0)) → c13(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c14(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c14(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c14(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, length(z0))) → c14(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, length1(z0))) → c14(CONS(proper(x0), length1(proper(z0))), PROPER(x0), PROPER(length1(z0)))
PROPER(cons(from(z0), x1)) → c14(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c14(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c14(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c14(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(length1(z0), x1)) → c14(CONS(length1(proper(z0)), proper(x1)), PROPER(length1(z0)), PROPER(x1))
PROPER(cons(x0, nil)) → c14(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, 0)) → c14(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(nil, x1)) → c14(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c14(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c15(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c15(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c15(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(length(z0))) → c15(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(length1(z0))) → c15(S(length1(proper(z0))), PROPER(length1(z0)))
PROPER(s(nil)) → c15(S(ok(nil)))
PROPER(s(0)) → c15(S(ok(0)))
PROPER(length(from(z0))) → c16(LENGTH(from(proper(z0))), PROPER(from(z0)))
PROPER(length(cons(z0, z1))) → c16(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(s(z0))) → c16(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(length(z0))) → c16(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(length1(z0))) → c16(LENGTH(length1(proper(z0))), PROPER(length1(z0)))
PROPER(length(nil)) → c16(LENGTH(ok(nil)))
PROPER(length(0)) → c16(LENGTH(ok(0)))
PROPER(length1(from(z0))) → c19(LENGTH1(from(proper(z0))), PROPER(from(z0)))
PROPER(length1(cons(z0, z1))) → c19(LENGTH1(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length1(s(z0))) → c19(LENGTH1(s(proper(z0))), PROPER(s(z0)))
PROPER(length1(length(z0))) → c19(LENGTH1(length(proper(z0))), PROPER(length(z0)))
PROPER(length1(nil)) → c19(LENGTH1(ok(nil)), PROPER(nil))
PROPER(length1(0)) → c19(LENGTH1(ok(0)), PROPER(0))
PROPER(length1(length1(z0))) → c19(LENGTH1(length1(proper(z0))), PROPER(length1(z0)))

K tuples:none
Defined Rule Symbols:

active, s, length1, from, cons, proper, length

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, LENGTH, LENGTH1, TOP, PROPER

Compound Symbols:

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

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

Removed 2 trailing tuple parts

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length1(z1)))
active(length1(z0)) → mark(length(z0))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
length1(ok(z0)) → ok(length1(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(0) → ok(0)
proper(length1(z0)) → length1(proper(z0))
length(ok(z0)) → ok(length(z0))

Tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(CONS(mark(s(length1(z1))), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c17(CONS(mark(length(z0)), x1))
ACTIVE(cons(length1(z0), x1)) → c17(ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(length(nil))) → c6(S(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c18(S(mark(s(length1(z1)))))
ACTIVE(s(length(cons(z0, z1)))) → c18(ACTIVE(length(cons(z0, z1))))
ACTIVE(s(length1(z0))) → c18(S(mark(length(z0))))
ACTIVE(s(length1(z0))) → c18(ACTIVE(length1(z0)))
PROPER(from(from(z0))) → c13(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c13(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c13(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(length(z0))) → c13(FROM(length(proper(z0))), PROPER(length(z0)))
PROPER(from(length1(z0))) → c13(FROM(length1(proper(z0))), PROPER(length1(z0)))
PROPER(from(nil)) → c13(FROM(ok(nil)))
PROPER(from(0)) → c13(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c14(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c14(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c14(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, length(z0))) → c14(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, length1(z0))) → c14(CONS(proper(x0), length1(proper(z0))), PROPER(x0), PROPER(length1(z0)))
PROPER(cons(from(z0), x1)) → c14(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c14(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c14(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c14(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(length1(z0), x1)) → c14(CONS(length1(proper(z0)), proper(x1)), PROPER(length1(z0)), PROPER(x1))
PROPER(cons(x0, nil)) → c14(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, 0)) → c14(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(nil, x1)) → c14(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c14(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c15(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c15(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c15(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(length(z0))) → c15(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(length1(z0))) → c15(S(length1(proper(z0))), PROPER(length1(z0)))
PROPER(s(nil)) → c15(S(ok(nil)))
PROPER(s(0)) → c15(S(ok(0)))
PROPER(length(from(z0))) → c16(LENGTH(from(proper(z0))), PROPER(from(z0)))
PROPER(length(cons(z0, z1))) → c16(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(s(z0))) → c16(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(length(z0))) → c16(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(length1(z0))) → c16(LENGTH(length1(proper(z0))), PROPER(length1(z0)))
PROPER(length(nil)) → c16(LENGTH(ok(nil)))
PROPER(length(0)) → c16(LENGTH(ok(0)))
PROPER(length1(from(z0))) → c19(LENGTH1(from(proper(z0))), PROPER(from(z0)))
PROPER(length1(cons(z0, z1))) → c19(LENGTH1(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length1(s(z0))) → c19(LENGTH1(s(proper(z0))), PROPER(s(z0)))
PROPER(length1(length(z0))) → c19(LENGTH1(length(proper(z0))), PROPER(length(z0)))
PROPER(length1(length1(z0))) → c19(LENGTH1(length1(proper(z0))), PROPER(length1(z0)))
PROPER(length1(nil)) → c19(LENGTH1(ok(nil)))
PROPER(length1(0)) → c19(LENGTH1(ok(0)))

S tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(mark(z0)) → c22(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(CONS(mark(s(length1(z1))), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c17(CONS(mark(length(z0)), x1))
ACTIVE(cons(length1(z0), x1)) → c17(ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(length(nil))) → c6(S(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c18(S(mark(s(length1(z1)))))
ACTIVE(s(length(cons(z0, z1)))) → c18(ACTIVE(length(cons(z0, z1))))
ACTIVE(s(length1(z0))) → c18(S(mark(length(z0))))
ACTIVE(s(length1(z0))) → c18(ACTIVE(length1(z0)))
PROPER(from(from(z0))) → c13(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c13(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c13(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(length(z0))) → c13(FROM(length(proper(z0))), PROPER(length(z0)))
PROPER(from(length1(z0))) → c13(FROM(length1(proper(z0))), PROPER(length1(z0)))
PROPER(from(nil)) → c13(FROM(ok(nil)))
PROPER(from(0)) → c13(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c14(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c14(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c14(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, length(z0))) → c14(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, length1(z0))) → c14(CONS(proper(x0), length1(proper(z0))), PROPER(x0), PROPER(length1(z0)))
PROPER(cons(from(z0), x1)) → c14(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c14(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c14(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c14(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(length1(z0), x1)) → c14(CONS(length1(proper(z0)), proper(x1)), PROPER(length1(z0)), PROPER(x1))
PROPER(cons(x0, nil)) → c14(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, 0)) → c14(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(nil, x1)) → c14(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c14(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c15(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c15(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c15(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(length(z0))) → c15(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(length1(z0))) → c15(S(length1(proper(z0))), PROPER(length1(z0)))
PROPER(s(nil)) → c15(S(ok(nil)))
PROPER(s(0)) → c15(S(ok(0)))
PROPER(length(from(z0))) → c16(LENGTH(from(proper(z0))), PROPER(from(z0)))
PROPER(length(cons(z0, z1))) → c16(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(s(z0))) → c16(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(length(z0))) → c16(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(length1(z0))) → c16(LENGTH(length1(proper(z0))), PROPER(length1(z0)))
PROPER(length(nil)) → c16(LENGTH(ok(nil)))
PROPER(length(0)) → c16(LENGTH(ok(0)))
PROPER(length1(from(z0))) → c19(LENGTH1(from(proper(z0))), PROPER(from(z0)))
PROPER(length1(cons(z0, z1))) → c19(LENGTH1(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length1(s(z0))) → c19(LENGTH1(s(proper(z0))), PROPER(s(z0)))
PROPER(length1(length(z0))) → c19(LENGTH1(length(proper(z0))), PROPER(length(z0)))
PROPER(length1(length1(z0))) → c19(LENGTH1(length1(proper(z0))), PROPER(length1(z0)))
PROPER(length1(nil)) → c19(LENGTH1(ok(nil)))
PROPER(length1(0)) → c19(LENGTH1(ok(0)))

K tuples:none
Defined Rule Symbols:

active, s, length1, from, cons, proper, length

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, LENGTH, LENGTH1, TOP, PROPER

Compound Symbols:

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

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

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

TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(length(z0))) → c22(TOP(length(proper(z0))), PROPER(length(z0)))
TOP(mark(nil)) → c22(TOP(ok(nil)), PROPER(nil))
TOP(mark(0)) → c22(TOP(ok(0)), PROPER(0))
TOP(mark(length1(z0))) → c22(TOP(length1(proper(z0))), PROPER(length1(z0)))

(50) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length1(z1)))
active(length1(z0)) → mark(length(z0))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
length1(ok(z0)) → ok(length1(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(0) → ok(0)
proper(length1(z0)) → length1(proper(z0))
length(ok(z0)) → ok(length(z0))

Tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(CONS(mark(s(length1(z1))), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c17(CONS(mark(length(z0)), x1))
ACTIVE(cons(length1(z0), x1)) → c17(ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(length(nil))) → c6(S(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c18(S(mark(s(length1(z1)))))
ACTIVE(s(length(cons(z0, z1)))) → c18(ACTIVE(length(cons(z0, z1))))
ACTIVE(s(length1(z0))) → c18(S(mark(length(z0))))
ACTIVE(s(length1(z0))) → c18(ACTIVE(length1(z0)))
PROPER(from(from(z0))) → c13(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c13(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c13(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(length(z0))) → c13(FROM(length(proper(z0))), PROPER(length(z0)))
PROPER(from(length1(z0))) → c13(FROM(length1(proper(z0))), PROPER(length1(z0)))
PROPER(from(nil)) → c13(FROM(ok(nil)))
PROPER(from(0)) → c13(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c14(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c14(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c14(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, length(z0))) → c14(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, length1(z0))) → c14(CONS(proper(x0), length1(proper(z0))), PROPER(x0), PROPER(length1(z0)))
PROPER(cons(from(z0), x1)) → c14(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c14(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c14(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c14(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(length1(z0), x1)) → c14(CONS(length1(proper(z0)), proper(x1)), PROPER(length1(z0)), PROPER(x1))
PROPER(cons(x0, nil)) → c14(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, 0)) → c14(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(nil, x1)) → c14(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c14(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c15(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c15(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c15(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(length(z0))) → c15(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(length1(z0))) → c15(S(length1(proper(z0))), PROPER(length1(z0)))
PROPER(s(nil)) → c15(S(ok(nil)))
PROPER(s(0)) → c15(S(ok(0)))
PROPER(length(from(z0))) → c16(LENGTH(from(proper(z0))), PROPER(from(z0)))
PROPER(length(cons(z0, z1))) → c16(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(s(z0))) → c16(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(length(z0))) → c16(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(length1(z0))) → c16(LENGTH(length1(proper(z0))), PROPER(length1(z0)))
PROPER(length(nil)) → c16(LENGTH(ok(nil)))
PROPER(length(0)) → c16(LENGTH(ok(0)))
PROPER(length1(from(z0))) → c19(LENGTH1(from(proper(z0))), PROPER(from(z0)))
PROPER(length1(cons(z0, z1))) → c19(LENGTH1(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length1(s(z0))) → c19(LENGTH1(s(proper(z0))), PROPER(s(z0)))
PROPER(length1(length(z0))) → c19(LENGTH1(length(proper(z0))), PROPER(length(z0)))
PROPER(length1(length1(z0))) → c19(LENGTH1(length1(proper(z0))), PROPER(length1(z0)))
PROPER(length1(nil)) → c19(LENGTH1(ok(nil)))
PROPER(length1(0)) → c19(LENGTH1(ok(0)))
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(length(z0))) → c22(TOP(length(proper(z0))), PROPER(length(z0)))
TOP(mark(nil)) → c22(TOP(ok(nil)), PROPER(nil))
TOP(mark(0)) → c22(TOP(ok(0)), PROPER(0))
TOP(mark(length1(z0))) → c22(TOP(length1(proper(z0))), PROPER(length1(z0)))

S tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(CONS(mark(s(length1(z1))), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c17(CONS(mark(length(z0)), x1))
ACTIVE(cons(length1(z0), x1)) → c17(ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(length(nil))) → c6(S(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c18(S(mark(s(length1(z1)))))
ACTIVE(s(length(cons(z0, z1)))) → c18(ACTIVE(length(cons(z0, z1))))
ACTIVE(s(length1(z0))) → c18(S(mark(length(z0))))
ACTIVE(s(length1(z0))) → c18(ACTIVE(length1(z0)))
PROPER(from(from(z0))) → c13(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c13(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c13(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(length(z0))) → c13(FROM(length(proper(z0))), PROPER(length(z0)))
PROPER(from(length1(z0))) → c13(FROM(length1(proper(z0))), PROPER(length1(z0)))
PROPER(from(nil)) → c13(FROM(ok(nil)))
PROPER(from(0)) → c13(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c14(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c14(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c14(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, length(z0))) → c14(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, length1(z0))) → c14(CONS(proper(x0), length1(proper(z0))), PROPER(x0), PROPER(length1(z0)))
PROPER(cons(from(z0), x1)) → c14(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c14(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c14(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c14(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(length1(z0), x1)) → c14(CONS(length1(proper(z0)), proper(x1)), PROPER(length1(z0)), PROPER(x1))
PROPER(cons(x0, nil)) → c14(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, 0)) → c14(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(nil, x1)) → c14(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c14(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c15(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c15(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c15(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(length(z0))) → c15(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(length1(z0))) → c15(S(length1(proper(z0))), PROPER(length1(z0)))
PROPER(s(nil)) → c15(S(ok(nil)))
PROPER(s(0)) → c15(S(ok(0)))
PROPER(length(from(z0))) → c16(LENGTH(from(proper(z0))), PROPER(from(z0)))
PROPER(length(cons(z0, z1))) → c16(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(s(z0))) → c16(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(length(z0))) → c16(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(length1(z0))) → c16(LENGTH(length1(proper(z0))), PROPER(length1(z0)))
PROPER(length(nil)) → c16(LENGTH(ok(nil)))
PROPER(length(0)) → c16(LENGTH(ok(0)))
PROPER(length1(from(z0))) → c19(LENGTH1(from(proper(z0))), PROPER(from(z0)))
PROPER(length1(cons(z0, z1))) → c19(LENGTH1(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length1(s(z0))) → c19(LENGTH1(s(proper(z0))), PROPER(s(z0)))
PROPER(length1(length(z0))) → c19(LENGTH1(length(proper(z0))), PROPER(length(z0)))
PROPER(length1(length1(z0))) → c19(LENGTH1(length1(proper(z0))), PROPER(length1(z0)))
PROPER(length1(nil)) → c19(LENGTH1(ok(nil)))
PROPER(length1(0)) → c19(LENGTH1(ok(0)))
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(length(z0))) → c22(TOP(length(proper(z0))), PROPER(length(z0)))
TOP(mark(nil)) → c22(TOP(ok(nil)), PROPER(nil))
TOP(mark(0)) → c22(TOP(ok(0)), PROPER(0))
TOP(mark(length1(z0))) → c22(TOP(length1(proper(z0))), PROPER(length1(z0)))

K tuples:none
Defined Rule Symbols:

active, s, length1, from, cons, proper, length

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, LENGTH, LENGTH1, TOP, PROPER

Compound Symbols:

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

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

Removed 2 trailing tuple parts

(52) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length1(z1)))
active(length1(z0)) → mark(length(z0))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
length1(ok(z0)) → ok(length1(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(0) → ok(0)
proper(length1(z0)) → length1(proper(z0))
length(ok(z0)) → ok(length(z0))

Tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(CONS(mark(s(length1(z1))), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c17(CONS(mark(length(z0)), x1))
ACTIVE(cons(length1(z0), x1)) → c17(ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(length(nil))) → c6(S(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c18(S(mark(s(length1(z1)))))
ACTIVE(s(length(cons(z0, z1)))) → c18(ACTIVE(length(cons(z0, z1))))
ACTIVE(s(length1(z0))) → c18(S(mark(length(z0))))
ACTIVE(s(length1(z0))) → c18(ACTIVE(length1(z0)))
PROPER(from(from(z0))) → c13(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c13(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c13(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(length(z0))) → c13(FROM(length(proper(z0))), PROPER(length(z0)))
PROPER(from(length1(z0))) → c13(FROM(length1(proper(z0))), PROPER(length1(z0)))
PROPER(from(nil)) → c13(FROM(ok(nil)))
PROPER(from(0)) → c13(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c14(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c14(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c14(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, length(z0))) → c14(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, length1(z0))) → c14(CONS(proper(x0), length1(proper(z0))), PROPER(x0), PROPER(length1(z0)))
PROPER(cons(from(z0), x1)) → c14(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c14(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c14(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c14(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(length1(z0), x1)) → c14(CONS(length1(proper(z0)), proper(x1)), PROPER(length1(z0)), PROPER(x1))
PROPER(cons(x0, nil)) → c14(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, 0)) → c14(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(nil, x1)) → c14(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c14(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c15(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c15(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c15(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(length(z0))) → c15(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(length1(z0))) → c15(S(length1(proper(z0))), PROPER(length1(z0)))
PROPER(s(nil)) → c15(S(ok(nil)))
PROPER(s(0)) → c15(S(ok(0)))
PROPER(length(from(z0))) → c16(LENGTH(from(proper(z0))), PROPER(from(z0)))
PROPER(length(cons(z0, z1))) → c16(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(s(z0))) → c16(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(length(z0))) → c16(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(length1(z0))) → c16(LENGTH(length1(proper(z0))), PROPER(length1(z0)))
PROPER(length(nil)) → c16(LENGTH(ok(nil)))
PROPER(length(0)) → c16(LENGTH(ok(0)))
PROPER(length1(from(z0))) → c19(LENGTH1(from(proper(z0))), PROPER(from(z0)))
PROPER(length1(cons(z0, z1))) → c19(LENGTH1(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length1(s(z0))) → c19(LENGTH1(s(proper(z0))), PROPER(s(z0)))
PROPER(length1(length(z0))) → c19(LENGTH1(length(proper(z0))), PROPER(length(z0)))
PROPER(length1(length1(z0))) → c19(LENGTH1(length1(proper(z0))), PROPER(length1(z0)))
PROPER(length1(nil)) → c19(LENGTH1(ok(nil)))
PROPER(length1(0)) → c19(LENGTH1(ok(0)))
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(length(z0))) → c22(TOP(length(proper(z0))), PROPER(length(z0)))
TOP(mark(length1(z0))) → c22(TOP(length1(proper(z0))), PROPER(length1(z0)))
TOP(mark(nil)) → c22(TOP(ok(nil)))
TOP(mark(0)) → c22(TOP(ok(0)))

S tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(CONS(mark(s(length1(z1))), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c17(CONS(mark(length(z0)), x1))
ACTIVE(cons(length1(z0), x1)) → c17(ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(length(nil))) → c6(S(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c18(S(mark(s(length1(z1)))))
ACTIVE(s(length(cons(z0, z1)))) → c18(ACTIVE(length(cons(z0, z1))))
ACTIVE(s(length1(z0))) → c18(S(mark(length(z0))))
ACTIVE(s(length1(z0))) → c18(ACTIVE(length1(z0)))
PROPER(from(from(z0))) → c13(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c13(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c13(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(length(z0))) → c13(FROM(length(proper(z0))), PROPER(length(z0)))
PROPER(from(length1(z0))) → c13(FROM(length1(proper(z0))), PROPER(length1(z0)))
PROPER(from(nil)) → c13(FROM(ok(nil)))
PROPER(from(0)) → c13(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c14(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c14(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c14(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, length(z0))) → c14(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, length1(z0))) → c14(CONS(proper(x0), length1(proper(z0))), PROPER(x0), PROPER(length1(z0)))
PROPER(cons(from(z0), x1)) → c14(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c14(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c14(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c14(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(length1(z0), x1)) → c14(CONS(length1(proper(z0)), proper(x1)), PROPER(length1(z0)), PROPER(x1))
PROPER(cons(x0, nil)) → c14(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, 0)) → c14(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(nil, x1)) → c14(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c14(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c15(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c15(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c15(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(length(z0))) → c15(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(length1(z0))) → c15(S(length1(proper(z0))), PROPER(length1(z0)))
PROPER(s(nil)) → c15(S(ok(nil)))
PROPER(s(0)) → c15(S(ok(0)))
PROPER(length(from(z0))) → c16(LENGTH(from(proper(z0))), PROPER(from(z0)))
PROPER(length(cons(z0, z1))) → c16(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(s(z0))) → c16(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(length(z0))) → c16(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(length1(z0))) → c16(LENGTH(length1(proper(z0))), PROPER(length1(z0)))
PROPER(length(nil)) → c16(LENGTH(ok(nil)))
PROPER(length(0)) → c16(LENGTH(ok(0)))
PROPER(length1(from(z0))) → c19(LENGTH1(from(proper(z0))), PROPER(from(z0)))
PROPER(length1(cons(z0, z1))) → c19(LENGTH1(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length1(s(z0))) → c19(LENGTH1(s(proper(z0))), PROPER(s(z0)))
PROPER(length1(length(z0))) → c19(LENGTH1(length(proper(z0))), PROPER(length(z0)))
PROPER(length1(length1(z0))) → c19(LENGTH1(length1(proper(z0))), PROPER(length1(z0)))
PROPER(length1(nil)) → c19(LENGTH1(ok(nil)))
PROPER(length1(0)) → c19(LENGTH1(ok(0)))
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(length(z0))) → c22(TOP(length(proper(z0))), PROPER(length(z0)))
TOP(mark(length1(z0))) → c22(TOP(length1(proper(z0))), PROPER(length1(z0)))
TOP(mark(nil)) → c22(TOP(ok(nil)))
TOP(mark(0)) → c22(TOP(ok(0)))

K tuples:none
Defined Rule Symbols:

active, s, length1, from, cons, proper, length

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, LENGTH, LENGTH1, TOP, PROPER

Compound Symbols:

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

(53) 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(nil)) → c22(TOP(ok(nil)))

We considered the (Usable) Rules:

active(cons(z0, z1)) → cons(active(z0), z1)
active(length(nil)) → mark(0)
s(mark(z0)) → mark(s(z0))
from(mark(z0)) → mark(from(z0))
active(length1(z0)) → mark(length(z0))
active(s(z0)) → s(active(z0))
active(length(cons(z0, z1))) → mark(s(length1(z1)))
active(from(z0)) → from(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
active(from(z0)) → mark(cons(z0, from(s(z0))))
length1(ok(z0)) → ok(length1(z0))
from(ok(z0)) → ok(from(z0))
length(ok(z0)) → ok(length(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
s(ok(z0)) → ok(s(z0))

And the Tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(CONS(mark(s(length1(z1))), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c17(CONS(mark(length(z0)), x1))
ACTIVE(cons(length1(z0), x1)) → c17(ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(length(nil))) → c6(S(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c18(S(mark(s(length1(z1)))))
ACTIVE(s(length(cons(z0, z1)))) → c18(ACTIVE(length(cons(z0, z1))))
ACTIVE(s(length1(z0))) → c18(S(mark(length(z0))))
ACTIVE(s(length1(z0))) → c18(ACTIVE(length1(z0)))
PROPER(from(from(z0))) → c13(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c13(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c13(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(length(z0))) → c13(FROM(length(proper(z0))), PROPER(length(z0)))
PROPER(from(length1(z0))) → c13(FROM(length1(proper(z0))), PROPER(length1(z0)))
PROPER(from(nil)) → c13(FROM(ok(nil)))
PROPER(from(0)) → c13(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c14(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c14(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c14(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, length(z0))) → c14(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, length1(z0))) → c14(CONS(proper(x0), length1(proper(z0))), PROPER(x0), PROPER(length1(z0)))
PROPER(cons(from(z0), x1)) → c14(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c14(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c14(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c14(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(length1(z0), x1)) → c14(CONS(length1(proper(z0)), proper(x1)), PROPER(length1(z0)), PROPER(x1))
PROPER(cons(x0, nil)) → c14(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, 0)) → c14(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(nil, x1)) → c14(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c14(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c15(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c15(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c15(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(length(z0))) → c15(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(length1(z0))) → c15(S(length1(proper(z0))), PROPER(length1(z0)))
PROPER(s(nil)) → c15(S(ok(nil)))
PROPER(s(0)) → c15(S(ok(0)))
PROPER(length(from(z0))) → c16(LENGTH(from(proper(z0))), PROPER(from(z0)))
PROPER(length(cons(z0, z1))) → c16(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(s(z0))) → c16(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(length(z0))) → c16(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(length1(z0))) → c16(LENGTH(length1(proper(z0))), PROPER(length1(z0)))
PROPER(length(nil)) → c16(LENGTH(ok(nil)))
PROPER(length(0)) → c16(LENGTH(ok(0)))
PROPER(length1(from(z0))) → c19(LENGTH1(from(proper(z0))), PROPER(from(z0)))
PROPER(length1(cons(z0, z1))) → c19(LENGTH1(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length1(s(z0))) → c19(LENGTH1(s(proper(z0))), PROPER(s(z0)))
PROPER(length1(length(z0))) → c19(LENGTH1(length(proper(z0))), PROPER(length(z0)))
PROPER(length1(length1(z0))) → c19(LENGTH1(length1(proper(z0))), PROPER(length1(z0)))
PROPER(length1(nil)) → c19(LENGTH1(ok(nil)))
PROPER(length1(0)) → c19(LENGTH1(ok(0)))
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(length(z0))) → c22(TOP(length(proper(z0))), PROPER(length(z0)))
TOP(mark(length1(z0))) → c22(TOP(length1(proper(z0))), PROPER(length1(z0)))
TOP(mark(nil)) → c22(TOP(ok(nil)))
TOP(mark(0)) → c22(TOP(ok(0)))

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

POL(0) = 0
POL(ACTIVE(x₁)) = 0
POL(CONS(x₁, x₂)) = 0
POL(FROM(x₁)) = 0
POL(LENGTH(x₁)) = 0
POL(LENGTH1(x₁)) = 0
POL(PROPER(x₁)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = [4]x₁
POL(active(x₁)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c13(x₁, x₂)) = x₁ + x₂
POL(c14(x₁, x₂)) = x₁ + x₂
POL(c14(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c15(x₁)) = x₁
POL(c15(x₁, x₂)) = x₁ + x₂
POL(c16(x₁)) = x₁
POL(c16(x₁, x₂)) = x₁ + x₂
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c19(x₁, x₂)) = x₁ + x₂
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c22(x₁, x₂)) = x₁ + x₂
POL(c23(x₁, x₂)) = x₁ + x₂
POL(c3(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c4(x₁, x₂)) = x₁ + x₂
POL(c5(x₁)) = x₁
POL(c5(x₁, x₂)) = x₁ + x₂
POL(c6(x₁)) = x₁
POL(c6(x₁, x₂)) = x₁ + x₂
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(cons(x₁, x₂)) = 0
POL(from(x₁)) = 0
POL(length(x₁)) = 0
POL(length1(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = [4]
POL(ok(x₁)) = 0
POL(proper(x₁)) = 0
POL(s(x₁)) = 0

(54) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length1(z1)))
active(length1(z0)) → mark(length(z0))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
length1(ok(z0)) → ok(length1(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(0) → ok(0)
proper(length1(z0)) → length1(proper(z0))
length(ok(z0)) → ok(length(z0))

Tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(CONS(mark(s(length1(z1))), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c17(CONS(mark(length(z0)), x1))
ACTIVE(cons(length1(z0), x1)) → c17(ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(length(nil))) → c6(S(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c18(S(mark(s(length1(z1)))))
ACTIVE(s(length(cons(z0, z1)))) → c18(ACTIVE(length(cons(z0, z1))))
ACTIVE(s(length1(z0))) → c18(S(mark(length(z0))))
ACTIVE(s(length1(z0))) → c18(ACTIVE(length1(z0)))
PROPER(from(from(z0))) → c13(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c13(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c13(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(length(z0))) → c13(FROM(length(proper(z0))), PROPER(length(z0)))
PROPER(from(length1(z0))) → c13(FROM(length1(proper(z0))), PROPER(length1(z0)))
PROPER(from(nil)) → c13(FROM(ok(nil)))
PROPER(from(0)) → c13(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c14(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c14(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c14(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, length(z0))) → c14(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, length1(z0))) → c14(CONS(proper(x0), length1(proper(z0))), PROPER(x0), PROPER(length1(z0)))
PROPER(cons(from(z0), x1)) → c14(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c14(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c14(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c14(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(length1(z0), x1)) → c14(CONS(length1(proper(z0)), proper(x1)), PROPER(length1(z0)), PROPER(x1))
PROPER(cons(x0, nil)) → c14(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, 0)) → c14(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(nil, x1)) → c14(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c14(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c15(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c15(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c15(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(length(z0))) → c15(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(length1(z0))) → c15(S(length1(proper(z0))), PROPER(length1(z0)))
PROPER(s(nil)) → c15(S(ok(nil)))
PROPER(s(0)) → c15(S(ok(0)))
PROPER(length(from(z0))) → c16(LENGTH(from(proper(z0))), PROPER(from(z0)))
PROPER(length(cons(z0, z1))) → c16(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(s(z0))) → c16(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(length(z0))) → c16(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(length1(z0))) → c16(LENGTH(length1(proper(z0))), PROPER(length1(z0)))
PROPER(length(nil)) → c16(LENGTH(ok(nil)))
PROPER(length(0)) → c16(LENGTH(ok(0)))
PROPER(length1(from(z0))) → c19(LENGTH1(from(proper(z0))), PROPER(from(z0)))
PROPER(length1(cons(z0, z1))) → c19(LENGTH1(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length1(s(z0))) → c19(LENGTH1(s(proper(z0))), PROPER(s(z0)))
PROPER(length1(length(z0))) → c19(LENGTH1(length(proper(z0))), PROPER(length(z0)))
PROPER(length1(length1(z0))) → c19(LENGTH1(length1(proper(z0))), PROPER(length1(z0)))
PROPER(length1(nil)) → c19(LENGTH1(ok(nil)))
PROPER(length1(0)) → c19(LENGTH1(ok(0)))
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(length(z0))) → c22(TOP(length(proper(z0))), PROPER(length(z0)))
TOP(mark(length1(z0))) → c22(TOP(length1(proper(z0))), PROPER(length1(z0)))
TOP(mark(nil)) → c22(TOP(ok(nil)))
TOP(mark(0)) → c22(TOP(ok(0)))

S tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(CONS(mark(s(length1(z1))), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c17(CONS(mark(length(z0)), x1))
ACTIVE(cons(length1(z0), x1)) → c17(ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(length(nil))) → c6(S(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c18(S(mark(s(length1(z1)))))
ACTIVE(s(length(cons(z0, z1)))) → c18(ACTIVE(length(cons(z0, z1))))
ACTIVE(s(length1(z0))) → c18(S(mark(length(z0))))
ACTIVE(s(length1(z0))) → c18(ACTIVE(length1(z0)))
PROPER(from(from(z0))) → c13(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c13(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c13(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(length(z0))) → c13(FROM(length(proper(z0))), PROPER(length(z0)))
PROPER(from(length1(z0))) → c13(FROM(length1(proper(z0))), PROPER(length1(z0)))
PROPER(from(nil)) → c13(FROM(ok(nil)))
PROPER(from(0)) → c13(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c14(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c14(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c14(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, length(z0))) → c14(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, length1(z0))) → c14(CONS(proper(x0), length1(proper(z0))), PROPER(x0), PROPER(length1(z0)))
PROPER(cons(from(z0), x1)) → c14(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c14(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c14(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c14(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(length1(z0), x1)) → c14(CONS(length1(proper(z0)), proper(x1)), PROPER(length1(z0)), PROPER(x1))
PROPER(cons(x0, nil)) → c14(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, 0)) → c14(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(nil, x1)) → c14(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c14(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c15(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c15(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c15(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(length(z0))) → c15(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(length1(z0))) → c15(S(length1(proper(z0))), PROPER(length1(z0)))
PROPER(s(nil)) → c15(S(ok(nil)))
PROPER(s(0)) → c15(S(ok(0)))
PROPER(length(from(z0))) → c16(LENGTH(from(proper(z0))), PROPER(from(z0)))
PROPER(length(cons(z0, z1))) → c16(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(s(z0))) → c16(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(length(z0))) → c16(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(length1(z0))) → c16(LENGTH(length1(proper(z0))), PROPER(length1(z0)))
PROPER(length(nil)) → c16(LENGTH(ok(nil)))
PROPER(length(0)) → c16(LENGTH(ok(0)))
PROPER(length1(from(z0))) → c19(LENGTH1(from(proper(z0))), PROPER(from(z0)))
PROPER(length1(cons(z0, z1))) → c19(LENGTH1(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length1(s(z0))) → c19(LENGTH1(s(proper(z0))), PROPER(s(z0)))
PROPER(length1(length(z0))) → c19(LENGTH1(length(proper(z0))), PROPER(length(z0)))
PROPER(length1(length1(z0))) → c19(LENGTH1(length1(proper(z0))), PROPER(length1(z0)))
PROPER(length1(nil)) → c19(LENGTH1(ok(nil)))
PROPER(length1(0)) → c19(LENGTH1(ok(0)))
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(length(z0))) → c22(TOP(length(proper(z0))), PROPER(length(z0)))
TOP(mark(length1(z0))) → c22(TOP(length1(proper(z0))), PROPER(length1(z0)))
TOP(mark(0)) → c22(TOP(ok(0)))

K tuples:

TOP(mark(nil)) → c22(TOP(ok(nil)))

Defined Rule Symbols:

active, s, length1, from, cons, proper, length

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, LENGTH, LENGTH1, TOP, PROPER

Compound Symbols:

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

(55) 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(0)) → c22(TOP(ok(0)))

We considered the (Usable) Rules:

active(cons(z0, z1)) → cons(active(z0), z1)
active(length(nil)) → mark(0)
s(mark(z0)) → mark(s(z0))
from(mark(z0)) → mark(from(z0))
active(length1(z0)) → mark(length(z0))
active(s(z0)) → s(active(z0))
active(length(cons(z0, z1))) → mark(s(length1(z1)))
active(from(z0)) → from(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
active(from(z0)) → mark(cons(z0, from(s(z0))))
length1(ok(z0)) → ok(length1(z0))
from(ok(z0)) → ok(from(z0))
length(ok(z0)) → ok(length(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
s(ok(z0)) → ok(s(z0))

And the Tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(CONS(mark(s(length1(z1))), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c17(CONS(mark(length(z0)), x1))
ACTIVE(cons(length1(z0), x1)) → c17(ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(length(nil))) → c6(S(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c18(S(mark(s(length1(z1)))))
ACTIVE(s(length(cons(z0, z1)))) → c18(ACTIVE(length(cons(z0, z1))))
ACTIVE(s(length1(z0))) → c18(S(mark(length(z0))))
ACTIVE(s(length1(z0))) → c18(ACTIVE(length1(z0)))
PROPER(from(from(z0))) → c13(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c13(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c13(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(length(z0))) → c13(FROM(length(proper(z0))), PROPER(length(z0)))
PROPER(from(length1(z0))) → c13(FROM(length1(proper(z0))), PROPER(length1(z0)))
PROPER(from(nil)) → c13(FROM(ok(nil)))
PROPER(from(0)) → c13(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c14(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c14(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c14(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, length(z0))) → c14(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, length1(z0))) → c14(CONS(proper(x0), length1(proper(z0))), PROPER(x0), PROPER(length1(z0)))
PROPER(cons(from(z0), x1)) → c14(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c14(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c14(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c14(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(length1(z0), x1)) → c14(CONS(length1(proper(z0)), proper(x1)), PROPER(length1(z0)), PROPER(x1))
PROPER(cons(x0, nil)) → c14(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, 0)) → c14(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(nil, x1)) → c14(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c14(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c15(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c15(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c15(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(length(z0))) → c15(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(length1(z0))) → c15(S(length1(proper(z0))), PROPER(length1(z0)))
PROPER(s(nil)) → c15(S(ok(nil)))
PROPER(s(0)) → c15(S(ok(0)))
PROPER(length(from(z0))) → c16(LENGTH(from(proper(z0))), PROPER(from(z0)))
PROPER(length(cons(z0, z1))) → c16(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(s(z0))) → c16(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(length(z0))) → c16(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(length1(z0))) → c16(LENGTH(length1(proper(z0))), PROPER(length1(z0)))
PROPER(length(nil)) → c16(LENGTH(ok(nil)))
PROPER(length(0)) → c16(LENGTH(ok(0)))
PROPER(length1(from(z0))) → c19(LENGTH1(from(proper(z0))), PROPER(from(z0)))
PROPER(length1(cons(z0, z1))) → c19(LENGTH1(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length1(s(z0))) → c19(LENGTH1(s(proper(z0))), PROPER(s(z0)))
PROPER(length1(length(z0))) → c19(LENGTH1(length(proper(z0))), PROPER(length(z0)))
PROPER(length1(length1(z0))) → c19(LENGTH1(length1(proper(z0))), PROPER(length1(z0)))
PROPER(length1(nil)) → c19(LENGTH1(ok(nil)))
PROPER(length1(0)) → c19(LENGTH1(ok(0)))
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(length(z0))) → c22(TOP(length(proper(z0))), PROPER(length(z0)))
TOP(mark(length1(z0))) → c22(TOP(length1(proper(z0))), PROPER(length1(z0)))
TOP(mark(nil)) → c22(TOP(ok(nil)))
TOP(mark(0)) → c22(TOP(ok(0)))

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

POL(0) = 0
POL(ACTIVE(x₁)) = 0
POL(CONS(x₁, x₂)) = 0
POL(FROM(x₁)) = 0
POL(LENGTH(x₁)) = 0
POL(LENGTH1(x₁)) = 0
POL(PROPER(x₁)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = [2]x₁
POL(active(x₁)) = x₁
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c13(x₁, x₂)) = x₁ + x₂
POL(c14(x₁, x₂)) = x₁ + x₂
POL(c14(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c15(x₁)) = x₁
POL(c15(x₁, x₂)) = x₁ + x₂
POL(c16(x₁)) = x₁
POL(c16(x₁, x₂)) = x₁ + x₂
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c19(x₁, x₂)) = x₁ + x₂
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c22(x₁, x₂)) = x₁ + x₂
POL(c23(x₁, x₂)) = x₁ + x₂
POL(c3(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c4(x₁, x₂)) = x₁ + x₂
POL(c5(x₁)) = x₁
POL(c5(x₁, x₂)) = x₁ + x₂
POL(c6(x₁)) = x₁
POL(c6(x₁, x₂)) = x₁ + x₂
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(cons(x₁, x₂)) = [1]
POL(from(x₁)) = [1]
POL(length(x₁)) = [1]
POL(length1(x₁)) = [1]
POL(mark(x₁)) = [1]
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(s(x₁)) = [1]

(56) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length1(z1)))
active(length1(z0)) → mark(length(z0))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
length1(ok(z0)) → ok(length1(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(0) → ok(0)
proper(length1(z0)) → length1(proper(z0))
length(ok(z0)) → ok(length(z0))

Tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(CONS(mark(s(length1(z1))), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c17(CONS(mark(length(z0)), x1))
ACTIVE(cons(length1(z0), x1)) → c17(ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(length(nil))) → c6(S(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c18(S(mark(s(length1(z1)))))
ACTIVE(s(length(cons(z0, z1)))) → c18(ACTIVE(length(cons(z0, z1))))
ACTIVE(s(length1(z0))) → c18(S(mark(length(z0))))
ACTIVE(s(length1(z0))) → c18(ACTIVE(length1(z0)))
PROPER(from(from(z0))) → c13(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c13(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c13(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(length(z0))) → c13(FROM(length(proper(z0))), PROPER(length(z0)))
PROPER(from(length1(z0))) → c13(FROM(length1(proper(z0))), PROPER(length1(z0)))
PROPER(from(nil)) → c13(FROM(ok(nil)))
PROPER(from(0)) → c13(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c14(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c14(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c14(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, length(z0))) → c14(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, length1(z0))) → c14(CONS(proper(x0), length1(proper(z0))), PROPER(x0), PROPER(length1(z0)))
PROPER(cons(from(z0), x1)) → c14(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c14(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c14(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c14(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(length1(z0), x1)) → c14(CONS(length1(proper(z0)), proper(x1)), PROPER(length1(z0)), PROPER(x1))
PROPER(cons(x0, nil)) → c14(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, 0)) → c14(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(nil, x1)) → c14(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c14(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c15(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c15(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c15(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(length(z0))) → c15(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(length1(z0))) → c15(S(length1(proper(z0))), PROPER(length1(z0)))
PROPER(s(nil)) → c15(S(ok(nil)))
PROPER(s(0)) → c15(S(ok(0)))
PROPER(length(from(z0))) → c16(LENGTH(from(proper(z0))), PROPER(from(z0)))
PROPER(length(cons(z0, z1))) → c16(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(s(z0))) → c16(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(length(z0))) → c16(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(length1(z0))) → c16(LENGTH(length1(proper(z0))), PROPER(length1(z0)))
PROPER(length(nil)) → c16(LENGTH(ok(nil)))
PROPER(length(0)) → c16(LENGTH(ok(0)))
PROPER(length1(from(z0))) → c19(LENGTH1(from(proper(z0))), PROPER(from(z0)))
PROPER(length1(cons(z0, z1))) → c19(LENGTH1(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length1(s(z0))) → c19(LENGTH1(s(proper(z0))), PROPER(s(z0)))
PROPER(length1(length(z0))) → c19(LENGTH1(length(proper(z0))), PROPER(length(z0)))
PROPER(length1(length1(z0))) → c19(LENGTH1(length1(proper(z0))), PROPER(length1(z0)))
PROPER(length1(nil)) → c19(LENGTH1(ok(nil)))
PROPER(length1(0)) → c19(LENGTH1(ok(0)))
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(length(z0))) → c22(TOP(length(proper(z0))), PROPER(length(z0)))
TOP(mark(length1(z0))) → c22(TOP(length1(proper(z0))), PROPER(length1(z0)))
TOP(mark(nil)) → c22(TOP(ok(nil)))
TOP(mark(0)) → c22(TOP(ok(0)))

S tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
TOP(ok(z0)) → c23(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(CONS(mark(s(length1(z1))), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c17(CONS(mark(length(z0)), x1))
ACTIVE(cons(length1(z0), x1)) → c17(ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(length(nil))) → c6(S(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c18(S(mark(s(length1(z1)))))
ACTIVE(s(length(cons(z0, z1)))) → c18(ACTIVE(length(cons(z0, z1))))
ACTIVE(s(length1(z0))) → c18(S(mark(length(z0))))
ACTIVE(s(length1(z0))) → c18(ACTIVE(length1(z0)))
PROPER(from(from(z0))) → c13(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c13(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c13(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(length(z0))) → c13(FROM(length(proper(z0))), PROPER(length(z0)))
PROPER(from(length1(z0))) → c13(FROM(length1(proper(z0))), PROPER(length1(z0)))
PROPER(from(nil)) → c13(FROM(ok(nil)))
PROPER(from(0)) → c13(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c14(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c14(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c14(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, length(z0))) → c14(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, length1(z0))) → c14(CONS(proper(x0), length1(proper(z0))), PROPER(x0), PROPER(length1(z0)))
PROPER(cons(from(z0), x1)) → c14(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c14(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c14(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c14(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(length1(z0), x1)) → c14(CONS(length1(proper(z0)), proper(x1)), PROPER(length1(z0)), PROPER(x1))
PROPER(cons(x0, nil)) → c14(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, 0)) → c14(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(nil, x1)) → c14(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c14(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c15(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c15(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c15(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(length(z0))) → c15(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(length1(z0))) → c15(S(length1(proper(z0))), PROPER(length1(z0)))
PROPER(s(nil)) → c15(S(ok(nil)))
PROPER(s(0)) → c15(S(ok(0)))
PROPER(length(from(z0))) → c16(LENGTH(from(proper(z0))), PROPER(from(z0)))
PROPER(length(cons(z0, z1))) → c16(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(s(z0))) → c16(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(length(z0))) → c16(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(length1(z0))) → c16(LENGTH(length1(proper(z0))), PROPER(length1(z0)))
PROPER(length(nil)) → c16(LENGTH(ok(nil)))
PROPER(length(0)) → c16(LENGTH(ok(0)))
PROPER(length1(from(z0))) → c19(LENGTH1(from(proper(z0))), PROPER(from(z0)))
PROPER(length1(cons(z0, z1))) → c19(LENGTH1(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length1(s(z0))) → c19(LENGTH1(s(proper(z0))), PROPER(s(z0)))
PROPER(length1(length(z0))) → c19(LENGTH1(length(proper(z0))), PROPER(length(z0)))
PROPER(length1(length1(z0))) → c19(LENGTH1(length1(proper(z0))), PROPER(length1(z0)))
PROPER(length1(nil)) → c19(LENGTH1(ok(nil)))
PROPER(length1(0)) → c19(LENGTH1(ok(0)))
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(length(z0))) → c22(TOP(length(proper(z0))), PROPER(length(z0)))
TOP(mark(length1(z0))) → c22(TOP(length1(proper(z0))), PROPER(length1(z0)))

K tuples:

TOP(mark(nil)) → c22(TOP(ok(nil)))
TOP(mark(0)) → c22(TOP(ok(0)))

Defined Rule Symbols:

active, s, length1, from, cons, proper, length

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, LENGTH, LENGTH1, TOP, PROPER

Compound Symbols:

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

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

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

TOP(ok(from(z0))) → c23(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(length(nil))) → c23(TOP(mark(0)), ACTIVE(length(nil)))
TOP(ok(length(cons(z0, z1)))) → c23(TOP(mark(s(length1(z1)))), ACTIVE(length(cons(z0, z1))))
TOP(ok(length1(z0))) → c23(TOP(mark(length(z0))), ACTIVE(length1(z0)))
TOP(ok(from(z0))) → c23(TOP(from(active(z0))), ACTIVE(from(z0)))
TOP(ok(cons(z0, z1))) → c23(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(s(z0))) → c23(TOP(s(active(z0))), ACTIVE(s(z0)))

(58) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length1(z1)))
active(length1(z0)) → mark(length(z0))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
length1(ok(z0)) → ok(length1(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(0) → ok(0)
proper(length1(z0)) → length1(proper(z0))
length(ok(z0)) → ok(length(z0))

Tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(CONS(mark(s(length1(z1))), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c17(CONS(mark(length(z0)), x1))
ACTIVE(cons(length1(z0), x1)) → c17(ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(length(nil))) → c6(S(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c18(S(mark(s(length1(z1)))))
ACTIVE(s(length(cons(z0, z1)))) → c18(ACTIVE(length(cons(z0, z1))))
ACTIVE(s(length1(z0))) → c18(S(mark(length(z0))))
ACTIVE(s(length1(z0))) → c18(ACTIVE(length1(z0)))
PROPER(from(from(z0))) → c13(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c13(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c13(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(length(z0))) → c13(FROM(length(proper(z0))), PROPER(length(z0)))
PROPER(from(length1(z0))) → c13(FROM(length1(proper(z0))), PROPER(length1(z0)))
PROPER(from(nil)) → c13(FROM(ok(nil)))
PROPER(from(0)) → c13(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c14(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c14(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c14(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, length(z0))) → c14(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, length1(z0))) → c14(CONS(proper(x0), length1(proper(z0))), PROPER(x0), PROPER(length1(z0)))
PROPER(cons(from(z0), x1)) → c14(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c14(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c14(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c14(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(length1(z0), x1)) → c14(CONS(length1(proper(z0)), proper(x1)), PROPER(length1(z0)), PROPER(x1))
PROPER(cons(x0, nil)) → c14(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, 0)) → c14(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(nil, x1)) → c14(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c14(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c15(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c15(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c15(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(length(z0))) → c15(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(length1(z0))) → c15(S(length1(proper(z0))), PROPER(length1(z0)))
PROPER(s(nil)) → c15(S(ok(nil)))
PROPER(s(0)) → c15(S(ok(0)))
PROPER(length(from(z0))) → c16(LENGTH(from(proper(z0))), PROPER(from(z0)))
PROPER(length(cons(z0, z1))) → c16(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(s(z0))) → c16(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(length(z0))) → c16(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(length1(z0))) → c16(LENGTH(length1(proper(z0))), PROPER(length1(z0)))
PROPER(length(nil)) → c16(LENGTH(ok(nil)))
PROPER(length(0)) → c16(LENGTH(ok(0)))
PROPER(length1(from(z0))) → c19(LENGTH1(from(proper(z0))), PROPER(from(z0)))
PROPER(length1(cons(z0, z1))) → c19(LENGTH1(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length1(s(z0))) → c19(LENGTH1(s(proper(z0))), PROPER(s(z0)))
PROPER(length1(length(z0))) → c19(LENGTH1(length(proper(z0))), PROPER(length(z0)))
PROPER(length1(length1(z0))) → c19(LENGTH1(length1(proper(z0))), PROPER(length1(z0)))
PROPER(length1(nil)) → c19(LENGTH1(ok(nil)))
PROPER(length1(0)) → c19(LENGTH1(ok(0)))
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(length(z0))) → c22(TOP(length(proper(z0))), PROPER(length(z0)))
TOP(mark(length1(z0))) → c22(TOP(length1(proper(z0))), PROPER(length1(z0)))
TOP(mark(nil)) → c22(TOP(ok(nil)))
TOP(mark(0)) → c22(TOP(ok(0)))
TOP(ok(from(z0))) → c23(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(length(nil))) → c23(TOP(mark(0)), ACTIVE(length(nil)))
TOP(ok(length(cons(z0, z1)))) → c23(TOP(mark(s(length1(z1)))), ACTIVE(length(cons(z0, z1))))
TOP(ok(length1(z0))) → c23(TOP(mark(length(z0))), ACTIVE(length1(z0)))
TOP(ok(from(z0))) → c23(TOP(from(active(z0))), ACTIVE(from(z0)))
TOP(ok(cons(z0, z1))) → c23(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(s(z0))) → c23(TOP(s(active(z0))), ACTIVE(s(z0)))

S tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(CONS(mark(s(length1(z1))), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c17(CONS(mark(length(z0)), x1))
ACTIVE(cons(length1(z0), x1)) → c17(ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(length(nil))) → c6(S(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c18(S(mark(s(length1(z1)))))
ACTIVE(s(length(cons(z0, z1)))) → c18(ACTIVE(length(cons(z0, z1))))
ACTIVE(s(length1(z0))) → c18(S(mark(length(z0))))
ACTIVE(s(length1(z0))) → c18(ACTIVE(length1(z0)))
PROPER(from(from(z0))) → c13(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c13(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c13(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(length(z0))) → c13(FROM(length(proper(z0))), PROPER(length(z0)))
PROPER(from(length1(z0))) → c13(FROM(length1(proper(z0))), PROPER(length1(z0)))
PROPER(from(nil)) → c13(FROM(ok(nil)))
PROPER(from(0)) → c13(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c14(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c14(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c14(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, length(z0))) → c14(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, length1(z0))) → c14(CONS(proper(x0), length1(proper(z0))), PROPER(x0), PROPER(length1(z0)))
PROPER(cons(from(z0), x1)) → c14(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c14(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c14(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c14(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(length1(z0), x1)) → c14(CONS(length1(proper(z0)), proper(x1)), PROPER(length1(z0)), PROPER(x1))
PROPER(cons(x0, nil)) → c14(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, 0)) → c14(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(nil, x1)) → c14(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c14(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c15(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c15(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c15(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(length(z0))) → c15(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(length1(z0))) → c15(S(length1(proper(z0))), PROPER(length1(z0)))
PROPER(s(nil)) → c15(S(ok(nil)))
PROPER(s(0)) → c15(S(ok(0)))
PROPER(length(from(z0))) → c16(LENGTH(from(proper(z0))), PROPER(from(z0)))
PROPER(length(cons(z0, z1))) → c16(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(s(z0))) → c16(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(length(z0))) → c16(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(length1(z0))) → c16(LENGTH(length1(proper(z0))), PROPER(length1(z0)))
PROPER(length(nil)) → c16(LENGTH(ok(nil)))
PROPER(length(0)) → c16(LENGTH(ok(0)))
PROPER(length1(from(z0))) → c19(LENGTH1(from(proper(z0))), PROPER(from(z0)))
PROPER(length1(cons(z0, z1))) → c19(LENGTH1(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length1(s(z0))) → c19(LENGTH1(s(proper(z0))), PROPER(s(z0)))
PROPER(length1(length(z0))) → c19(LENGTH1(length(proper(z0))), PROPER(length(z0)))
PROPER(length1(length1(z0))) → c19(LENGTH1(length1(proper(z0))), PROPER(length1(z0)))
PROPER(length1(nil)) → c19(LENGTH1(ok(nil)))
PROPER(length1(0)) → c19(LENGTH1(ok(0)))
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(length(z0))) → c22(TOP(length(proper(z0))), PROPER(length(z0)))
TOP(mark(length1(z0))) → c22(TOP(length1(proper(z0))), PROPER(length1(z0)))
TOP(ok(from(z0))) → c23(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(length(nil))) → c23(TOP(mark(0)), ACTIVE(length(nil)))
TOP(ok(length(cons(z0, z1)))) → c23(TOP(mark(s(length1(z1)))), ACTIVE(length(cons(z0, z1))))
TOP(ok(length1(z0))) → c23(TOP(mark(length(z0))), ACTIVE(length1(z0)))
TOP(ok(from(z0))) → c23(TOP(from(active(z0))), ACTIVE(from(z0)))
TOP(ok(cons(z0, z1))) → c23(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(s(z0))) → c23(TOP(s(active(z0))), ACTIVE(s(z0)))

K tuples:

TOP(mark(nil)) → c22(TOP(ok(nil)))
TOP(mark(0)) → c22(TOP(ok(0)))

Defined Rule Symbols:

active, s, length1, from, cons, proper, length

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, LENGTH, LENGTH1, PROPER, TOP

Compound Symbols:

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

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

Removed 3 trailing nodes:

TOP(mark(nil)) → c22(TOP(ok(nil)))
TOP(ok(length(nil))) → c23(TOP(mark(0)), ACTIVE(length(nil)))
TOP(mark(0)) → c22(TOP(ok(0)))

(60) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length1(z1)))
active(length1(z0)) → mark(length(z0))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
length1(ok(z0)) → ok(length1(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(0) → ok(0)
proper(length1(z0)) → length1(proper(z0))
length(ok(z0)) → ok(length(z0))

Tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(CONS(mark(s(length1(z1))), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c17(CONS(mark(length(z0)), x1))
ACTIVE(cons(length1(z0), x1)) → c17(ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(length(nil))) → c6(S(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c18(S(mark(s(length1(z1)))))
ACTIVE(s(length(cons(z0, z1)))) → c18(ACTIVE(length(cons(z0, z1))))
ACTIVE(s(length1(z0))) → c18(S(mark(length(z0))))
ACTIVE(s(length1(z0))) → c18(ACTIVE(length1(z0)))
PROPER(from(from(z0))) → c13(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c13(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c13(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(length(z0))) → c13(FROM(length(proper(z0))), PROPER(length(z0)))
PROPER(from(length1(z0))) → c13(FROM(length1(proper(z0))), PROPER(length1(z0)))
PROPER(from(nil)) → c13(FROM(ok(nil)))
PROPER(from(0)) → c13(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c14(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c14(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c14(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, length(z0))) → c14(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, length1(z0))) → c14(CONS(proper(x0), length1(proper(z0))), PROPER(x0), PROPER(length1(z0)))
PROPER(cons(from(z0), x1)) → c14(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c14(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c14(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c14(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(length1(z0), x1)) → c14(CONS(length1(proper(z0)), proper(x1)), PROPER(length1(z0)), PROPER(x1))
PROPER(cons(x0, nil)) → c14(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, 0)) → c14(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(nil, x1)) → c14(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c14(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c15(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c15(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c15(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(length(z0))) → c15(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(length1(z0))) → c15(S(length1(proper(z0))), PROPER(length1(z0)))
PROPER(s(nil)) → c15(S(ok(nil)))
PROPER(s(0)) → c15(S(ok(0)))
PROPER(length(from(z0))) → c16(LENGTH(from(proper(z0))), PROPER(from(z0)))
PROPER(length(cons(z0, z1))) → c16(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(s(z0))) → c16(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(length(z0))) → c16(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(length1(z0))) → c16(LENGTH(length1(proper(z0))), PROPER(length1(z0)))
PROPER(length(nil)) → c16(LENGTH(ok(nil)))
PROPER(length(0)) → c16(LENGTH(ok(0)))
PROPER(length1(from(z0))) → c19(LENGTH1(from(proper(z0))), PROPER(from(z0)))
PROPER(length1(cons(z0, z1))) → c19(LENGTH1(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length1(s(z0))) → c19(LENGTH1(s(proper(z0))), PROPER(s(z0)))
PROPER(length1(length(z0))) → c19(LENGTH1(length(proper(z0))), PROPER(length(z0)))
PROPER(length1(length1(z0))) → c19(LENGTH1(length1(proper(z0))), PROPER(length1(z0)))
PROPER(length1(nil)) → c19(LENGTH1(ok(nil)))
PROPER(length1(0)) → c19(LENGTH1(ok(0)))
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(length(z0))) → c22(TOP(length(proper(z0))), PROPER(length(z0)))
TOP(mark(length1(z0))) → c22(TOP(length1(proper(z0))), PROPER(length1(z0)))
TOP(ok(from(z0))) → c23(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(length(cons(z0, z1)))) → c23(TOP(mark(s(length1(z1)))), ACTIVE(length(cons(z0, z1))))
TOP(ok(length1(z0))) → c23(TOP(mark(length(z0))), ACTIVE(length1(z0)))
TOP(ok(from(z0))) → c23(TOP(from(active(z0))), ACTIVE(from(z0)))
TOP(ok(cons(z0, z1))) → c23(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(s(z0))) → c23(TOP(s(active(z0))), ACTIVE(s(z0)))

S tuples:

ACTIVE(length1(z0)) → c3(LENGTH(z0))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(CONS(mark(s(length1(z1))), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c17(CONS(mark(length(z0)), x1))
ACTIVE(cons(length1(z0), x1)) → c17(ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(length(nil))) → c6(S(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c18(S(mark(s(length1(z1)))))
ACTIVE(s(length(cons(z0, z1)))) → c18(ACTIVE(length(cons(z0, z1))))
ACTIVE(s(length1(z0))) → c18(S(mark(length(z0))))
ACTIVE(s(length1(z0))) → c18(ACTIVE(length1(z0)))
PROPER(from(from(z0))) → c13(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c13(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c13(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(length(z0))) → c13(FROM(length(proper(z0))), PROPER(length(z0)))
PROPER(from(length1(z0))) → c13(FROM(length1(proper(z0))), PROPER(length1(z0)))
PROPER(from(nil)) → c13(FROM(ok(nil)))
PROPER(from(0)) → c13(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c14(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c14(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c14(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, length(z0))) → c14(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, length1(z0))) → c14(CONS(proper(x0), length1(proper(z0))), PROPER(x0), PROPER(length1(z0)))
PROPER(cons(from(z0), x1)) → c14(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c14(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c14(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c14(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(length1(z0), x1)) → c14(CONS(length1(proper(z0)), proper(x1)), PROPER(length1(z0)), PROPER(x1))
PROPER(cons(x0, nil)) → c14(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, 0)) → c14(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(nil, x1)) → c14(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c14(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c15(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c15(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c15(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(length(z0))) → c15(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(length1(z0))) → c15(S(length1(proper(z0))), PROPER(length1(z0)))
PROPER(s(nil)) → c15(S(ok(nil)))
PROPER(s(0)) → c15(S(ok(0)))
PROPER(length(from(z0))) → c16(LENGTH(from(proper(z0))), PROPER(from(z0)))
PROPER(length(cons(z0, z1))) → c16(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(s(z0))) → c16(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(length(z0))) → c16(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(length1(z0))) → c16(LENGTH(length1(proper(z0))), PROPER(length1(z0)))
PROPER(length(nil)) → c16(LENGTH(ok(nil)))
PROPER(length(0)) → c16(LENGTH(ok(0)))
PROPER(length1(from(z0))) → c19(LENGTH1(from(proper(z0))), PROPER(from(z0)))
PROPER(length1(cons(z0, z1))) → c19(LENGTH1(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length1(s(z0))) → c19(LENGTH1(s(proper(z0))), PROPER(s(z0)))
PROPER(length1(length(z0))) → c19(LENGTH1(length(proper(z0))), PROPER(length(z0)))
PROPER(length1(length1(z0))) → c19(LENGTH1(length1(proper(z0))), PROPER(length1(z0)))
PROPER(length1(nil)) → c19(LENGTH1(ok(nil)))
PROPER(length1(0)) → c19(LENGTH1(ok(0)))
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(length(z0))) → c22(TOP(length(proper(z0))), PROPER(length(z0)))
TOP(mark(length1(z0))) → c22(TOP(length1(proper(z0))), PROPER(length1(z0)))
TOP(ok(from(z0))) → c23(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(length(cons(z0, z1)))) → c23(TOP(mark(s(length1(z1)))), ACTIVE(length(cons(z0, z1))))
TOP(ok(length1(z0))) → c23(TOP(mark(length(z0))), ACTIVE(length1(z0)))
TOP(ok(from(z0))) → c23(TOP(from(active(z0))), ACTIVE(from(z0)))
TOP(ok(cons(z0, z1))) → c23(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(s(z0))) → c23(TOP(s(active(z0))), ACTIVE(s(z0)))

K tuples:none
Defined Rule Symbols:

active, s, length1, from, cons, proper, length

Defined Pair Symbols:

ACTIVE, FROM, CONS, S, LENGTH, LENGTH1, PROPER, TOP

Compound Symbols:

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

(61) CdtUnreachableProof (EQUIVALENT transformation)

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

ACTIVE(length1(z0)) → c3(LENGTH(z0))
ACTIVE(from(z0)) → c(S(z0))
ACTIVE(length(cons(z0, z1))) → c1(S(length1(z1)))
ACTIVE(length(cons(z0, z1))) → c1(LENGTH1(z1))
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(from(length(nil))) → c4(FROM(mark(0)))
ACTIVE(from(length(cons(z0, z1)))) → c2(FROM(mark(s(length1(z1)))))
ACTIVE(from(length(cons(z0, z1)))) → c2(ACTIVE(length(cons(z0, z1))))
ACTIVE(from(length1(z0))) → c2(FROM(mark(length(z0))))
ACTIVE(from(length1(z0))) → c2(ACTIVE(length1(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0)))
ACTIVE(cons(from(z0), x1)) → c5(CONS(from(active(z0)), x1), ACTIVE(from(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c5(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(s(z0), x1)) → c5(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(length(nil), x1)) → c5(CONS(mark(0), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(CONS(mark(s(length1(z1))), x1))
ACTIVE(cons(length(cons(z0, z1)), x1)) → c17(ACTIVE(length(cons(z0, z1))))
ACTIVE(cons(length1(z0), x1)) → c17(CONS(mark(length(z0)), x1))
ACTIVE(cons(length1(z0), x1)) → c17(ACTIVE(length1(z0)))
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0)))
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(length(nil))) → c6(S(mark(0)))
ACTIVE(s(length(cons(z0, z1)))) → c18(S(mark(s(length1(z1)))))
ACTIVE(s(length(cons(z0, z1)))) → c18(ACTIVE(length(cons(z0, z1))))
ACTIVE(s(length1(z0))) → c18(S(mark(length(z0))))
ACTIVE(s(length1(z0))) → c18(ACTIVE(length1(z0)))
PROPER(from(from(z0))) → c13(FROM(from(proper(z0))), PROPER(from(z0)))
PROPER(from(cons(z0, z1))) → c13(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(from(s(z0))) → c13(FROM(s(proper(z0))), PROPER(s(z0)))
PROPER(from(length(z0))) → c13(FROM(length(proper(z0))), PROPER(length(z0)))
PROPER(from(length1(z0))) → c13(FROM(length1(proper(z0))), PROPER(length1(z0)))
PROPER(from(nil)) → c13(FROM(ok(nil)))
PROPER(from(0)) → c13(FROM(ok(0)))
PROPER(cons(x0, from(z0))) → c14(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0)))
PROPER(cons(x0, cons(z0, z1))) → c14(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, s(z0))) → c14(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, length(z0))) → c14(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(x0, length1(z0))) → c14(CONS(proper(x0), length1(proper(z0))), PROPER(x0), PROPER(length1(z0)))
PROPER(cons(from(z0), x1)) → c14(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c14(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c14(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c14(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(length1(z0), x1)) → c14(CONS(length1(proper(z0)), proper(x1)), PROPER(length1(z0)), PROPER(x1))
PROPER(cons(x0, nil)) → c14(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(x0, 0)) → c14(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(nil, x1)) → c14(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(cons(0, x1)) → c14(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(from(z0))) → c15(S(from(proper(z0))), PROPER(from(z0)))
PROPER(s(cons(z0, z1))) → c15(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(s(z0))) → c15(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(length(z0))) → c15(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(length1(z0))) → c15(S(length1(proper(z0))), PROPER(length1(z0)))
PROPER(s(nil)) → c15(S(ok(nil)))
PROPER(s(0)) → c15(S(ok(0)))
PROPER(length(from(z0))) → c16(LENGTH(from(proper(z0))), PROPER(from(z0)))
PROPER(length(cons(z0, z1))) → c16(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(s(z0))) → c16(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(length(z0))) → c16(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(length1(z0))) → c16(LENGTH(length1(proper(z0))), PROPER(length1(z0)))
PROPER(length(nil)) → c16(LENGTH(ok(nil)))
PROPER(length(0)) → c16(LENGTH(ok(0)))
PROPER(length1(from(z0))) → c19(LENGTH1(from(proper(z0))), PROPER(from(z0)))
PROPER(length1(cons(z0, z1))) → c19(LENGTH1(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length1(s(z0))) → c19(LENGTH1(s(proper(z0))), PROPER(s(z0)))
PROPER(length1(length(z0))) → c19(LENGTH1(length(proper(z0))), PROPER(length(z0)))
PROPER(length1(length1(z0))) → c19(LENGTH1(length1(proper(z0))), PROPER(length1(z0)))
PROPER(length1(nil)) → c19(LENGTH1(ok(nil)))
PROPER(length1(0)) → c19(LENGTH1(ok(0)))
TOP(mark(from(z0))) → c22(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(cons(z0, z1))) → c22(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c22(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(length(z0))) → c22(TOP(length(proper(z0))), PROPER(length(z0)))
TOP(mark(length1(z0))) → c22(TOP(length1(proper(z0))), PROPER(length1(z0)))
TOP(ok(from(z0))) → c23(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(length(cons(z0, z1)))) → c23(TOP(mark(s(length1(z1)))), ACTIVE(length(cons(z0, z1))))
TOP(ok(length1(z0))) → c23(TOP(mark(length(z0))), ACTIVE(length1(z0)))
TOP(ok(from(z0))) → c23(TOP(from(active(z0))), ACTIVE(from(z0)))
TOP(ok(cons(z0, z1))) → c23(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(s(z0))) → c23(TOP(s(active(z0))), ACTIVE(s(z0)))

(62) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length1(z1)))
active(length1(z0)) → mark(length(z0))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
length1(ok(z0)) → ok(length1(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(0) → ok(0)
proper(length1(z0)) → length1(proper(z0))
length(ok(z0)) → ok(length(z0))

Tuples:

FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))

S tuples:

FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))

K tuples:none
Defined Rule Symbols:

active, s, length1, from, cons, proper, length

Defined Pair Symbols:

FROM, CONS, S, LENGTH, LENGTH1

Compound Symbols:

c7, c8, c9, c10, c11, c12, c20, c21

(63) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

active(from(z0)) → mark(cons(z0, from(s(z0))))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length1(z1)))
active(length1(z0)) → mark(length(z0))
active(from(z0)) → from(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(s(z0)) → s(active(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
length1(ok(z0)) → ok(length1(z0))
from(mark(z0)) → mark(from(z0))
from(ok(z0)) → ok(from(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
proper(from(z0)) → from(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(s(z0)) → s(proper(z0))
proper(length(z0)) → length(proper(z0))
proper(nil) → ok(nil)
proper(0) → ok(0)
proper(length1(z0)) → length1(proper(z0))
length(ok(z0)) → ok(length(z0))

(64) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))

S tuples:

FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))

K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

FROM, CONS, S, LENGTH, LENGTH1

Compound Symbols:

c7, c8, c9, c10, c11, c12, c20, c21

(65) 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.

LENGTH(ok(z0)) → c20(LENGTH(z0))

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

FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))

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

POL(CONS(x₁, x₂)) = 0
POL(FROM(x₁)) = 0
POL(LENGTH(x₁)) = x₁
POL(LENGTH1(x₁)) = 0
POL(S(x₁)) = 0
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = [4]
POL(ok(x₁)) = [2] + x₁

(66) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))

S tuples:

FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))

K tuples:

LENGTH(ok(z0)) → c20(LENGTH(z0))

Defined Rule Symbols:none

Defined Pair Symbols:

FROM, CONS, S, LENGTH, LENGTH1

Compound Symbols:

c7, c8, c9, c10, c11, c12, c20, c21

(67) 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))

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

FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))

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

POL(CONS(x₁, x₂)) = 0
POL(FROM(x₁)) = 0
POL(LENGTH(x₁)) = [2]x₁
POL(LENGTH1(x₁)) = 0
POL(S(x₁)) = [2]x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = [4] + x₁
POL(ok(x₁)) = x₁

(68) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))

S tuples:

FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(ok(z0)) → c12(S(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))

K tuples:

LENGTH(ok(z0)) → c20(LENGTH(z0))
S(mark(z0)) → c11(S(z0))

Defined Rule Symbols:none

Defined Pair Symbols:

FROM, CONS, S, LENGTH, LENGTH1

Compound Symbols:

c7, c8, c9, c10, c11, c12, c20, c21

(69) 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.

CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))

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

FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))

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

POL(CONS(x₁, x₂)) = [2]x₂
POL(FROM(x₁)) = 0
POL(LENGTH(x₁)) = 0
POL(LENGTH1(x₁)) = 0
POL(S(x₁)) = 0
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = [2]
POL(ok(x₁)) = [4] + x₁

(70) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))

S tuples:

FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
S(ok(z0)) → c12(S(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))

K tuples:

LENGTH(ok(z0)) → c20(LENGTH(z0))
S(mark(z0)) → c11(S(z0))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))

Defined Rule Symbols:none

Defined Pair Symbols:

FROM, CONS, S, LENGTH, LENGTH1

Compound Symbols:

c7, c8, c9, c10, c11, c12, c20, c21

(71) 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.

FROM(mark(z0)) → c7(FROM(z0))

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

FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))

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

POL(CONS(x₁, x₂)) = 0
POL(FROM(x₁)) = [2]x₁
POL(LENGTH(x₁)) = [4]x₁
POL(LENGTH1(x₁)) = 0
POL(S(x₁)) = [3]x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = [2] + x₁
POL(ok(x₁)) = x₁

(72) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))

S tuples:

FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
S(ok(z0)) → c12(S(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))

K tuples:

LENGTH(ok(z0)) → c20(LENGTH(z0))
S(mark(z0)) → c11(S(z0))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
FROM(mark(z0)) → c7(FROM(z0))

Defined Rule Symbols:none

Defined Pair Symbols:

FROM, CONS, S, LENGTH, LENGTH1

Compound Symbols:

c7, c8, c9, c10, c11, c12, c20, c21

(73) 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.

FROM(ok(z0)) → c8(FROM(z0))

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

FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))

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

POL(CONS(x₁, x₂)) = [3]x₂
POL(FROM(x₁)) = [4]x₁
POL(LENGTH(x₁)) = [4]x₁
POL(LENGTH1(x₁)) = 0
POL(S(x₁)) = 0
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = [3] + x₁
POL(ok(x₁)) = [5] + x₁

(74) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))

S tuples:

CONS(mark(z0), z1) → c9(CONS(z0, z1))
S(ok(z0)) → c12(S(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))

K tuples:

LENGTH(ok(z0)) → c20(LENGTH(z0))
S(mark(z0)) → c11(S(z0))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))

Defined Rule Symbols:none

Defined Pair Symbols:

FROM, CONS, S, LENGTH, LENGTH1

Compound Symbols:

c7, c8, c9, c10, c11, c12, c20, c21

(75) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

LENGTH1(ok(z0)) → c21(LENGTH1(z0))

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

FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))

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

POL(CONS(x₁, x₂)) = x₂²
POL(FROM(x₁)) = 0
POL(LENGTH(x₁)) = [2]x₁²
POL(LENGTH1(x₁)) = [2]x₁²
POL(S(x₁)) = 0
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = 0
POL(ok(x₁)) = [1] + x₁

(76) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))

S tuples:

CONS(mark(z0), z1) → c9(CONS(z0, z1))
S(ok(z0)) → c12(S(z0))

K tuples:

LENGTH(ok(z0)) → c20(LENGTH(z0))
S(mark(z0)) → c11(S(z0))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))

Defined Rule Symbols:none

Defined Pair Symbols:

FROM, CONS, S, LENGTH, LENGTH1

Compound Symbols:

c7, c8, c9, c10, c11, c12, c20, c21

(77) 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.

CONS(mark(z0), z1) → c9(CONS(z0, z1))

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

FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))

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

POL(CONS(x₁, x₂)) = x₁ + [3]x₂
POL(FROM(x₁)) = 0
POL(LENGTH(x₁)) = [4]x₁
POL(LENGTH1(x₁)) = [3]x₁
POL(S(x₁)) = 0
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁

(78) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))

S tuples:

S(ok(z0)) → c12(S(z0))

K tuples:

LENGTH(ok(z0)) → c20(LENGTH(z0))
S(mark(z0)) → c11(S(z0))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))

Defined Rule Symbols:none

Defined Pair Symbols:

FROM, CONS, S, LENGTH, LENGTH1

Compound Symbols:

c7, c8, c9, c10, c11, c12, c20, c21

(79) 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(ok(z0)) → c12(S(z0))

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

FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))

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

POL(CONS(x₁, x₂)) = [4]x₁ + [3]x₂
POL(FROM(x₁)) = [4]x₁
POL(LENGTH(x₁)) = 0
POL(LENGTH1(x₁)) = [3]x₁
POL(S(x₁)) = [4]x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(ok(x₁)) = [1] + x₁

(80) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
S(mark(z0)) → c11(S(z0))
S(ok(z0)) → c12(S(z0))
LENGTH(ok(z0)) → c20(LENGTH(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))

S tuples:none
K tuples:

LENGTH(ok(z0)) → c20(LENGTH(z0))
S(mark(z0)) → c11(S(z0))
CONS(ok(z0), ok(z1)) → c10(CONS(z0, z1))
FROM(mark(z0)) → c7(FROM(z0))
FROM(ok(z0)) → c8(FROM(z0))
LENGTH1(ok(z0)) → c21(LENGTH1(z0))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
S(ok(z0)) → c12(S(z0))

Defined Rule Symbols:none

Defined Pair Symbols:

FROM, CONS, S, LENGTH, LENGTH1

Compound Symbols:

c7, c8, c9, c10, c11, c12, c20, c21

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

The set S is empty