Runtime Complexity (innermost) proof of /tmp/tmpb55aji/Ex1_GL02a

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

0 CpxTRS

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

↳2 CdtProblem

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

↳4 CdtProblem

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

↳6 CdtProblem

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

↳8 CdtProblem

↳9 CdtUsableRulesProof (⇔, 85 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), 0 ms)

↳22 CdtProblem

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

↳24 CdtProblem

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

↳26 CdtProblem

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

↳28 CdtProblem

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

↳30 CdtProblem

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

↳32 CdtProblem

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

↳34 CdtProblem

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

↳36 CdtProblem

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

↳38 CdtProblem

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

↳40 CdtProblem

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

↳42 CdtProblem

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

↳44 CdtProblem

↳45 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 2 ms)

↳46 CdtProblem

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

↳48 CdtProblem

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

↳50 CdtProblem

↳51 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 2069 ms)

↳52 CdtProblem

↳53 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 1428 ms)

↳54 CdtProblem

↳55 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 2198 ms)

↳56 CdtProblem

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

↳58 CdtProblem

↳59 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 7 ms)

↳60 CdtProblem

↳61 CdtUnreachableProof (⇔, 0 ms)

↳62 CdtProblem

↳63 CdtUsableRulesProof (⇔, 0 ms)

↳64 CdtProblem

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

↳76 CdtProblem

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

↳78 BOUNDS(O(1), O(1))

(0) Obligation:

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

active(eq(0, 0)) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0, X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0)
active(length(cons(X, L))) → mark(s(length(L)))
active(inf(X)) → inf(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(length(X)) → length(active(X))
inf(mark(X)) → mark(inf(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
length(mark(X)) → mark(length(X))
proper(eq(X1, X2)) → eq(proper(X1), proper(X2))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(X)) → s(proper(X))
proper(false) → ok(false)
proper(inf(X)) → inf(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(length(X)) → length(proper(X))
eq(ok(X1), ok(X2)) → ok(eq(X1, X2))
s(ok(X)) → ok(s(X))
inf(ok(X)) → ok(inf(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
length(ok(X)) → ok(length(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(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(eq(0, 0)) → c
ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(eq(z0, z1)) → c2
ACTIVE(inf(z0)) → c3(CONS(z0, inf(s(z0))), INF(s(z0)), S(z0))
ACTIVE(take(0, z0)) → c4
ACTIVE(take(s(z0), cons(z1, z2))) → c5(CONS(z1, take(z0, z2)), TAKE(z0, z2))
ACTIVE(length(nil)) → c6
ACTIVE(length(cons(z0, z1))) → c7(S(length(z1)), LENGTH(z1))
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(0) → c20
PROPER(true) → c21
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(false) → c23
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(nil) → c27
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(eq(0, 0)) → c
ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(eq(z0, z1)) → c2
ACTIVE(inf(z0)) → c3(CONS(z0, inf(s(z0))), INF(s(z0)), S(z0))
ACTIVE(take(0, z0)) → c4
ACTIVE(take(s(z0), cons(z1, z2))) → c5(CONS(z1, take(z0, z2)), TAKE(z0, z2))
ACTIVE(length(nil)) → c6
ACTIVE(length(cons(z0, z1))) → c7(S(length(z1)), LENGTH(z1))
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(0) → c20
PROPER(true) → c21
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(false) → c23
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(nil) → c27
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, inf, take, length, proper, eq, s, cons, top

Defined Pair Symbols:

ACTIVE, INF, TAKE, LENGTH, PROPER, EQ, S, CONS, TOP

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33

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

Removed 8 trailing nodes:

ACTIVE(eq(0, 0)) → c
PROPER(true) → c21
PROPER(false) → c23
ACTIVE(eq(z0, z1)) → c2
ACTIVE(take(0, z0)) → c4
ACTIVE(length(nil)) → c6
PROPER(nil) → c27
PROPER(0) → c20

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(inf(z0)) → c3(CONS(z0, inf(s(z0))), INF(s(z0)), S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(CONS(z1, take(z0, z2)), TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c7(S(length(z1)), LENGTH(z1))
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(inf(z0)) → c3(CONS(z0, inf(s(z0))), INF(s(z0)), S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(CONS(z1, take(z0, z2)), TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c7(S(length(z1)), LENGTH(z1))
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, inf, take, length, proper, eq, s, cons, top

Defined Pair Symbols:

ACTIVE, INF, TAKE, LENGTH, PROPER, EQ, S, CONS, TOP

Compound Symbols:

c1, c3, c5, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c22, c24, c25, c26, c28, c29, c30, c31, c32, c33

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

Removed 2 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(CONS(z1, take(z0, z2)), TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c7(S(length(z1)), LENGTH(z1))
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(s(z0), cons(z1, z2))) → c5(CONS(z1, take(z0, z2)), TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c7(S(length(z1)), LENGTH(z1))
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))

K tuples:none
Defined Rule Symbols:

active, inf, take, length, proper, eq, s, cons, top

Defined Pair Symbols:

ACTIVE, INF, TAKE, LENGTH, PROPER, EQ, S, CONS, TOP

Compound Symbols:

c1, c5, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c22, c24, c25, c26, c28, c29, c30, c31, c32, c33, c3

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

Split RHS of tuples not part of any SCC

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))
s(ok(z0)) → ok(s(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))

K tuples:none
Defined Rule Symbols:

active, inf, take, length, proper, eq, s, cons, top

Defined Pair Symbols:

ACTIVE, INF, TAKE, LENGTH, PROPER, EQ, S, CONS, TOP

Compound Symbols:

c1, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c22, c24, c25, c26, c28, c29, c30, c31, c32, c33, c3, c

(9) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

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

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
s(ok(z0)) → ok(s(z0))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))

K tuples:none
Defined Rule Symbols:

active, cons, take, s, length, inf, proper, eq

Defined Pair Symbols:

ACTIVE, INF, TAKE, LENGTH, PROPER, EQ, S, CONS, TOP

Compound Symbols:

c1, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c22, c24, c25, c26, c28, c29, c30, c31, c32, c33, c3, c

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

Use narrowing to replace ACTIVE(inf(z0)) → c8(INF(active(z0)), ACTIVE(z0)) by

ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)), ACTIVE(eq(0, 0)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c8(INF(mark(eq(z0, z1))), ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c8(INF(mark(false)), ACTIVE(eq(z0, z1)))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)), ACTIVE(length(nil)))
ACTIVE(inf(length(cons(z0, z1)))) → c8(INF(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
s(ok(z0)) → ok(s(z0))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)), ACTIVE(eq(0, 0)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c8(INF(mark(eq(z0, z1))), ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c8(INF(mark(false)), ACTIVE(eq(z0, z1)))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)), ACTIVE(length(nil)))
ACTIVE(inf(length(cons(z0, z1)))) → c8(INF(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)), ACTIVE(eq(0, 0)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c8(INF(mark(eq(z0, z1))), ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c8(INF(mark(false)), ACTIVE(eq(z0, z1)))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)), ACTIVE(length(nil)))
ACTIVE(inf(length(cons(z0, z1)))) → c8(INF(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))

K tuples:none
Defined Rule Symbols:

active, cons, take, s, length, inf, proper, eq

Defined Pair Symbols:

ACTIVE, INF, TAKE, LENGTH, PROPER, EQ, S, CONS, TOP

Compound Symbols:

c1, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c22, c24, c25, c26, c28, c29, c30, c31, c32, c33, c3, c, c8

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

Removed 1 trailing tuple parts

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
s(ok(z0)) → ok(s(z0))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(eq(s(z0), s(z1)))) → c8(INF(mark(eq(z0, z1))), ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c8(INF(mark(false)), ACTIVE(eq(z0, z1)))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)), ACTIVE(length(nil)))
ACTIVE(inf(length(cons(z0, z1)))) → c8(INF(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(eq(s(z0), s(z1)))) → c8(INF(mark(eq(z0, z1))), ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c8(INF(mark(false)), ACTIVE(eq(z0, z1)))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)), ACTIVE(length(nil)))
ACTIVE(inf(length(cons(z0, z1)))) → c8(INF(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))

K tuples:none
Defined Rule Symbols:

active, cons, take, s, length, inf, proper, eq

Defined Pair Symbols:

ACTIVE, INF, TAKE, LENGTH, PROPER, EQ, S, CONS, TOP

Compound Symbols:

c1, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c22, c24, c25, c26, c28, c29, c30, c31, c32, c33, c3, c, c8, c8

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

Split RHS of tuples not part of any SCC

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
s(ok(z0)) → ok(s(z0))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)), ACTIVE(length(nil)))
ACTIVE(inf(length(cons(z0, z1)))) → c8(INF(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)), ACTIVE(length(nil)))
ACTIVE(inf(length(cons(z0, z1)))) → c8(INF(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))

K tuples:none
Defined Rule Symbols:

active, cons, take, s, length, inf, proper, eq

Defined Pair Symbols:

ACTIVE, INF, TAKE, LENGTH, PROPER, EQ, S, CONS, TOP

Compound Symbols:

c1, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c22, c24, c25, c26, c28, c29, c30, c31, c32, c33, c3, c, c8, c8, c2

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

Use narrowing to replace ACTIVE(length(z0)) → c11(LENGTH(active(z0)), ACTIVE(z0)) by

ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)), ACTIVE(eq(0, 0)))
ACTIVE(length(eq(s(z0), s(z1)))) → c11(LENGTH(mark(eq(z0, z1))), ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c11(LENGTH(mark(false)), ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)), ACTIVE(length(nil)))
ACTIVE(length(length(cons(z0, z1)))) → c11(LENGTH(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
s(ok(z0)) → ok(s(z0))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)), ACTIVE(length(nil)))
ACTIVE(inf(length(cons(z0, z1)))) → c8(INF(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)), ACTIVE(eq(0, 0)))
ACTIVE(length(eq(s(z0), s(z1)))) → c11(LENGTH(mark(eq(z0, z1))), ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c11(LENGTH(mark(false)), ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)), ACTIVE(length(nil)))
ACTIVE(length(length(cons(z0, z1)))) → c11(LENGTH(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)), ACTIVE(length(nil)))
ACTIVE(inf(length(cons(z0, z1)))) → c8(INF(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)), ACTIVE(eq(0, 0)))
ACTIVE(length(eq(s(z0), s(z1)))) → c11(LENGTH(mark(eq(z0, z1))), ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c11(LENGTH(mark(false)), ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)), ACTIVE(length(nil)))
ACTIVE(length(length(cons(z0, z1)))) → c11(LENGTH(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))

K tuples:none
Defined Rule Symbols:

active, cons, take, s, length, inf, proper, eq

Defined Pair Symbols:

ACTIVE, INF, TAKE, LENGTH, PROPER, EQ, S, CONS, TOP

Compound Symbols:

c1, c9, c10, c12, c13, c14, c15, c16, c17, c18, c19, c22, c24, c25, c26, c28, c29, c30, c31, c32, c33, c3, c, c8, c8, c2, c11

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

Removed 3 trailing tuple parts

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
s(ok(z0)) → ok(s(z0))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(length(cons(z0, z1)))) → c8(INF(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(eq(s(z0), s(z1)))) → c11(LENGTH(mark(eq(z0, z1))), ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c11(LENGTH(mark(false)), ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(length(cons(z0, z1)))) → c11(LENGTH(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(length(cons(z0, z1)))) → c8(INF(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(eq(s(z0), s(z1)))) → c11(LENGTH(mark(eq(z0, z1))), ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c11(LENGTH(mark(false)), ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(length(cons(z0, z1)))) → c11(LENGTH(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))

K tuples:none
Defined Rule Symbols:

active, cons, take, s, length, inf, proper, eq

Defined Pair Symbols:

ACTIVE, INF, TAKE, LENGTH, PROPER, EQ, S, CONS, TOP

Compound Symbols:

c1, c9, c10, c12, c13, c14, c15, c16, c17, c18, c19, c22, c24, c25, c26, c28, c29, c30, c31, c32, c33, c3, c, c8, c8, c2, c11, c11

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

Split RHS of tuples not part of any SCC

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
s(ok(z0)) → ok(s(z0))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))

K tuples:none
Defined Rule Symbols:

active, cons, take, s, length, inf, proper, eq

Defined Pair Symbols:

ACTIVE, INF, TAKE, LENGTH, PROPER, EQ, S, CONS, TOP

Compound Symbols:

c1, c9, c10, c12, c13, c14, c15, c16, c17, c18, c19, c22, c24, c25, c26, c28, c29, c30, c31, c32, c33, c3, c, c8, c8, c2, c11, c11, c4

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

Use narrowing to replace PROPER(eq(z0, z1)) → c19(EQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0), PROPER(true))
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0), PROPER(false))
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(true), PROPER(x1))
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(false), PROPER(x1))
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
s(ok(z0)) → ok(s(z0))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0), PROPER(true))
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0), PROPER(false))
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(true), PROPER(x1))
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(false), PROPER(x1))
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0), PROPER(true))
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0), PROPER(false))
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(true), PROPER(x1))
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(false), PROPER(x1))
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, cons, take, s, length, inf, proper, eq

Defined Pair Symbols:

ACTIVE, INF, TAKE, LENGTH, PROPER, EQ, S, CONS, TOP

Compound Symbols:

c1, c9, c10, c12, c13, c14, c15, c16, c17, c18, c22, c24, c25, c26, c28, c29, c30, c31, c32, c33, c3, c, c8, c8, c2, c11, c11, c4, c19

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

Removed 8 trailing tuple parts

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
s(ok(z0)) → ok(s(z0))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0))
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0))
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0))
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0))
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1))
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1))
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1))
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0))
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0))
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0))
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0))
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1))
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1))
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1))
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, cons, take, s, length, inf, proper, eq

Defined Pair Symbols:

ACTIVE, INF, TAKE, LENGTH, PROPER, EQ, S, CONS, TOP

Compound Symbols:

c1, c9, c10, c12, c13, c14, c15, c16, c17, c18, c22, c24, c25, c26, c28, c29, c30, c31, c32, c33, c3, c, c8, c8, c2, c11, c11, c4, c19, c19

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

Use narrowing to replace PROPER(s(z0)) → c22(S(proper(z0)), PROPER(z0)) by

PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(s(0)) → c22(S(ok(0)), PROPER(0))
PROPER(s(true)) → c22(S(ok(true)), PROPER(true))
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(false)) → c22(S(ok(false)), PROPER(false))
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0)))
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(s(nil)) → c22(S(ok(nil)), PROPER(nil))
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0)))

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
s(ok(z0)) → ok(s(z0))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0))
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0))
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0))
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0))
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1))
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1))
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1))
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(s(0)) → c22(S(ok(0)), PROPER(0))
PROPER(s(true)) → c22(S(ok(true)), PROPER(true))
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(false)) → c22(S(ok(false)), PROPER(false))
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0)))
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(s(nil)) → c22(S(ok(nil)), PROPER(nil))
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0)))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0))
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0))
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0))
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0))
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1))
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1))
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1))
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(s(0)) → c22(S(ok(0)), PROPER(0))
PROPER(s(true)) → c22(S(ok(true)), PROPER(true))
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(false)) → c22(S(ok(false)), PROPER(false))
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0)))
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(s(nil)) → c22(S(ok(nil)), PROPER(nil))
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0)))

K tuples:none
Defined Rule Symbols:

active, cons, take, s, length, inf, proper, eq

Defined Pair Symbols:

ACTIVE, INF, TAKE, LENGTH, PROPER, EQ, S, CONS, TOP

Compound Symbols:

c1, c9, c10, c12, c13, c14, c15, c16, c17, c18, c24, c25, c26, c28, c29, c30, c31, c32, c33, c3, c, c8, c8, c2, c11, c11, c4, c19, c19, c22

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

Removed 4 trailing tuple parts

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
s(ok(z0)) → ok(s(z0))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0))
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0))
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0))
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0))
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1))
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1))
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1))
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0)))
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(0)) → c22(S(ok(0)))
PROPER(s(true)) → c22(S(ok(true)))
PROPER(s(false)) → c22(S(ok(false)))
PROPER(s(nil)) → c22(S(ok(nil)))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0))
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0))
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0))
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0))
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1))
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1))
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1))
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0)))
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(0)) → c22(S(ok(0)))
PROPER(s(true)) → c22(S(ok(true)))
PROPER(s(false)) → c22(S(ok(false)))
PROPER(s(nil)) → c22(S(ok(nil)))

K tuples:none
Defined Rule Symbols:

active, cons, take, s, length, inf, proper, eq

Defined Pair Symbols:

ACTIVE, INF, TAKE, LENGTH, PROPER, EQ, S, CONS, TOP

Compound Symbols:

c1, c9, c10, c12, c13, c14, c15, c16, c17, c18, c24, c25, c26, c28, c29, c30, c31, c32, c33, c3, c, c8, c8, c2, c11, c11, c4, c19, c19, c22, c22

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

Use narrowing to replace PROPER(inf(z0)) → c24(INF(proper(z0)), PROPER(z0)) by

PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(inf(0)) → c24(INF(ok(0)), PROPER(0))
PROPER(inf(true)) → c24(INF(ok(true)), PROPER(true))
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0)))
PROPER(inf(false)) → c24(INF(ok(false)), PROPER(false))
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0)))
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(inf(nil)) → c24(INF(ok(nil)), PROPER(nil))
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0)))

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
s(ok(z0)) → ok(s(z0))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0))
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0))
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0))
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0))
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1))
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1))
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1))
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0)))
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(0)) → c22(S(ok(0)))
PROPER(s(true)) → c22(S(ok(true)))
PROPER(s(false)) → c22(S(ok(false)))
PROPER(s(nil)) → c22(S(ok(nil)))
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(inf(0)) → c24(INF(ok(0)), PROPER(0))
PROPER(inf(true)) → c24(INF(ok(true)), PROPER(true))
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0)))
PROPER(inf(false)) → c24(INF(ok(false)), PROPER(false))
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0)))
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(inf(nil)) → c24(INF(ok(nil)), PROPER(nil))
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0)))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0))
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0))
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0))
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0))
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1))
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1))
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1))
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0)))
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(0)) → c22(S(ok(0)))
PROPER(s(true)) → c22(S(ok(true)))
PROPER(s(false)) → c22(S(ok(false)))
PROPER(s(nil)) → c22(S(ok(nil)))
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(inf(0)) → c24(INF(ok(0)), PROPER(0))
PROPER(inf(true)) → c24(INF(ok(true)), PROPER(true))
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0)))
PROPER(inf(false)) → c24(INF(ok(false)), PROPER(false))
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0)))
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(inf(nil)) → c24(INF(ok(nil)), PROPER(nil))
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0)))

K tuples:none
Defined Rule Symbols:

active, cons, take, s, length, inf, proper, eq

Defined Pair Symbols:

ACTIVE, INF, TAKE, LENGTH, PROPER, EQ, S, CONS, TOP

Compound Symbols:

c1, c9, c10, c12, c13, c14, c15, c16, c17, c18, c25, c26, c28, c29, c30, c31, c32, c33, c3, c, c8, c8, c2, c11, c11, c4, c19, c19, c22, c22, c24

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

Removed 4 trailing tuple parts

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
s(ok(z0)) → ok(s(z0))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0))
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0))
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0))
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0))
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1))
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1))
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1))
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0)))
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(0)) → c22(S(ok(0)))
PROPER(s(true)) → c22(S(ok(true)))
PROPER(s(false)) → c22(S(ok(false)))
PROPER(s(nil)) → c22(S(ok(nil)))
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0)))
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0)))
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0)))
PROPER(inf(0)) → c24(INF(ok(0)))
PROPER(inf(true)) → c24(INF(ok(true)))
PROPER(inf(false)) → c24(INF(ok(false)))
PROPER(inf(nil)) → c24(INF(ok(nil)))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0))
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0))
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0))
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0))
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1))
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1))
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1))
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0)))
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(0)) → c22(S(ok(0)))
PROPER(s(true)) → c22(S(ok(true)))
PROPER(s(false)) → c22(S(ok(false)))
PROPER(s(nil)) → c22(S(ok(nil)))
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0)))
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0)))
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0)))
PROPER(inf(0)) → c24(INF(ok(0)))
PROPER(inf(true)) → c24(INF(ok(true)))
PROPER(inf(false)) → c24(INF(ok(false)))
PROPER(inf(nil)) → c24(INF(ok(nil)))

K tuples:none
Defined Rule Symbols:

active, cons, take, s, length, inf, proper, eq

Defined Pair Symbols:

ACTIVE, INF, TAKE, LENGTH, PROPER, EQ, S, CONS, TOP

Compound Symbols:

c1, c9, c10, c12, c13, c14, c15, c16, c17, c18, c25, c26, c28, c29, c30, c31, c32, c33, c3, c, c8, c8, c2, c11, c11, c4, c19, c19, c22, c22, c24, c24

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

Use narrowing to replace PROPER(cons(z0, z1)) → c25(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0), PROPER(true))
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0), PROPER(false))
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(true), PROPER(x1))
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(false), PROPER(x1))
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
s(ok(z0)) → ok(s(z0))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0))
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0))
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0))
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0))
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1))
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1))
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1))
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0)))
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(0)) → c22(S(ok(0)))
PROPER(s(true)) → c22(S(ok(true)))
PROPER(s(false)) → c22(S(ok(false)))
PROPER(s(nil)) → c22(S(ok(nil)))
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0)))
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0)))
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0)))
PROPER(inf(0)) → c24(INF(ok(0)))
PROPER(inf(true)) → c24(INF(ok(true)))
PROPER(inf(false)) → c24(INF(ok(false)))
PROPER(inf(nil)) → c24(INF(ok(nil)))
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0), PROPER(true))
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0), PROPER(false))
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(true), PROPER(x1))
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(false), PROPER(x1))
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0))
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0))
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0))
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0))
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1))
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1))
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1))
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0)))
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(0)) → c22(S(ok(0)))
PROPER(s(true)) → c22(S(ok(true)))
PROPER(s(false)) → c22(S(ok(false)))
PROPER(s(nil)) → c22(S(ok(nil)))
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0)))
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0)))
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0)))
PROPER(inf(0)) → c24(INF(ok(0)))
PROPER(inf(true)) → c24(INF(ok(true)))
PROPER(inf(false)) → c24(INF(ok(false)))
PROPER(inf(nil)) → c24(INF(ok(nil)))
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0), PROPER(true))
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0), PROPER(false))
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(true), PROPER(x1))
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(false), PROPER(x1))
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, cons, take, s, length, inf, proper, eq

Defined Pair Symbols:

ACTIVE, INF, TAKE, LENGTH, PROPER, EQ, S, CONS, TOP

Compound Symbols:

c1, c9, c10, c12, c13, c14, c15, c16, c17, c18, c26, c28, c29, c30, c31, c32, c33, c3, c, c8, c8, c2, c11, c11, c4, c19, c19, c22, c22, c24, c24, c25

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

Removed 8 trailing tuple parts

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
s(ok(z0)) → ok(s(z0))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0))
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0))
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0))
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0))
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1))
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1))
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1))
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0)))
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(0)) → c22(S(ok(0)))
PROPER(s(true)) → c22(S(ok(true)))
PROPER(s(false)) → c22(S(ok(false)))
PROPER(s(nil)) → c22(S(ok(nil)))
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0)))
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0)))
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0)))
PROPER(inf(0)) → c24(INF(ok(0)))
PROPER(inf(true)) → c24(INF(ok(true)))
PROPER(inf(false)) → c24(INF(ok(false)))
PROPER(inf(nil)) → c24(INF(ok(nil)))
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0))
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0))
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1))
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0))
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0))
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0))
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0))
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1))
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1))
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1))
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0)))
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(0)) → c22(S(ok(0)))
PROPER(s(true)) → c22(S(ok(true)))
PROPER(s(false)) → c22(S(ok(false)))
PROPER(s(nil)) → c22(S(ok(nil)))
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0)))
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0)))
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0)))
PROPER(inf(0)) → c24(INF(ok(0)))
PROPER(inf(true)) → c24(INF(ok(true)))
PROPER(inf(false)) → c24(INF(ok(false)))
PROPER(inf(nil)) → c24(INF(ok(nil)))
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0))
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0))
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1))
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, cons, take, s, length, inf, proper, eq

Defined Pair Symbols:

ACTIVE, INF, TAKE, LENGTH, PROPER, EQ, S, CONS, TOP

Compound Symbols:

c1, c9, c10, c12, c13, c14, c15, c16, c17, c18, c26, c28, c29, c30, c31, c32, c33, c3, c, c8, c8, c2, c11, c11, c4, c19, c19, c22, c22, c24, c24, c25, c25

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

Use narrowing to replace PROPER(take(z0, z1)) → c26(TAKE(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0), PROPER(true))
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0), PROPER(false))
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(true), PROPER(x1))
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(false), PROPER(x1))
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
s(ok(z0)) → ok(s(z0))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0))
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0))
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0))
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0))
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1))
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1))
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1))
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0)))
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(0)) → c22(S(ok(0)))
PROPER(s(true)) → c22(S(ok(true)))
PROPER(s(false)) → c22(S(ok(false)))
PROPER(s(nil)) → c22(S(ok(nil)))
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0)))
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0)))
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0)))
PROPER(inf(0)) → c24(INF(ok(0)))
PROPER(inf(true)) → c24(INF(ok(true)))
PROPER(inf(false)) → c24(INF(ok(false)))
PROPER(inf(nil)) → c24(INF(ok(nil)))
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0))
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0))
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1))
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0), PROPER(true))
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0), PROPER(false))
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(true), PROPER(x1))
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(false), PROPER(x1))
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0))
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0))
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0))
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0))
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1))
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1))
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1))
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0)))
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(0)) → c22(S(ok(0)))
PROPER(s(true)) → c22(S(ok(true)))
PROPER(s(false)) → c22(S(ok(false)))
PROPER(s(nil)) → c22(S(ok(nil)))
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0)))
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0)))
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0)))
PROPER(inf(0)) → c24(INF(ok(0)))
PROPER(inf(true)) → c24(INF(ok(true)))
PROPER(inf(false)) → c24(INF(ok(false)))
PROPER(inf(nil)) → c24(INF(ok(nil)))
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0))
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0))
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1))
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0), PROPER(true))
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0), PROPER(false))
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(true), PROPER(x1))
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(false), PROPER(x1))
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, cons, take, s, length, inf, proper, eq

Defined Pair Symbols:

ACTIVE, INF, TAKE, LENGTH, PROPER, EQ, S, CONS, TOP

Compound Symbols:

c1, c9, c10, c12, c13, c14, c15, c16, c17, c18, c28, c29, c30, c31, c32, c33, c3, c, c8, c8, c2, c11, c11, c4, c19, c19, c22, c22, c24, c24, c25, c25, c26

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

Removed 8 trailing tuple parts

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
s(ok(z0)) → ok(s(z0))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0))
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0))
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0))
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0))
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1))
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1))
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1))
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0)))
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(0)) → c22(S(ok(0)))
PROPER(s(true)) → c22(S(ok(true)))
PROPER(s(false)) → c22(S(ok(false)))
PROPER(s(nil)) → c22(S(ok(nil)))
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0)))
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0)))
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0)))
PROPER(inf(0)) → c24(INF(ok(0)))
PROPER(inf(true)) → c24(INF(ok(true)))
PROPER(inf(false)) → c24(INF(ok(false)))
PROPER(inf(nil)) → c24(INF(ok(nil)))
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0))
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0))
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1))
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0))
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0))
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0))
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0))
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(x1))
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(x1))
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(x1))
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(x1))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0))
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0))
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0))
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0))
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1))
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1))
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1))
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0)))
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(0)) → c22(S(ok(0)))
PROPER(s(true)) → c22(S(ok(true)))
PROPER(s(false)) → c22(S(ok(false)))
PROPER(s(nil)) → c22(S(ok(nil)))
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0)))
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0)))
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0)))
PROPER(inf(0)) → c24(INF(ok(0)))
PROPER(inf(true)) → c24(INF(ok(true)))
PROPER(inf(false)) → c24(INF(ok(false)))
PROPER(inf(nil)) → c24(INF(ok(nil)))
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0))
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0))
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1))
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0))
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0))
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0))
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0))
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(x1))
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(x1))
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(x1))
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, cons, take, s, length, inf, proper, eq

Defined Pair Symbols:

ACTIVE, INF, TAKE, LENGTH, PROPER, EQ, S, CONS, TOP

Compound Symbols:

c1, c9, c10, c12, c13, c14, c15, c16, c17, c18, c28, c29, c30, c31, c32, c33, c3, c, c8, c8, c2, c11, c11, c4, c19, c19, c22, c22, c24, c24, c25, c25, c26, c26

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

Use narrowing to replace PROPER(length(z0)) → c28(LENGTH(proper(z0)), PROPER(z0)) by

PROPER(length(eq(z0, z1))) → c28(LENGTH(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(length(0)) → c28(LENGTH(ok(0)), PROPER(0))
PROPER(length(true)) → c28(LENGTH(ok(true)), PROPER(true))
PROPER(length(s(z0))) → c28(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(false)) → c28(LENGTH(ok(false)), PROPER(false))
PROPER(length(inf(z0))) → c28(LENGTH(inf(proper(z0))), PROPER(inf(z0)))
PROPER(length(cons(z0, z1))) → c28(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(take(z0, z1))) → c28(LENGTH(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(length(nil)) → c28(LENGTH(ok(nil)), PROPER(nil))
PROPER(length(length(z0))) → c28(LENGTH(length(proper(z0))), PROPER(length(z0)))

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
s(ok(z0)) → ok(s(z0))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0))
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0))
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0))
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0))
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1))
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1))
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1))
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0)))
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(0)) → c22(S(ok(0)))
PROPER(s(true)) → c22(S(ok(true)))
PROPER(s(false)) → c22(S(ok(false)))
PROPER(s(nil)) → c22(S(ok(nil)))
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0)))
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0)))
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0)))
PROPER(inf(0)) → c24(INF(ok(0)))
PROPER(inf(true)) → c24(INF(ok(true)))
PROPER(inf(false)) → c24(INF(ok(false)))
PROPER(inf(nil)) → c24(INF(ok(nil)))
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0))
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0))
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1))
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0))
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0))
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0))
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0))
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(x1))
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(x1))
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(x1))
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(x1))
PROPER(length(eq(z0, z1))) → c28(LENGTH(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(length(0)) → c28(LENGTH(ok(0)), PROPER(0))
PROPER(length(true)) → c28(LENGTH(ok(true)), PROPER(true))
PROPER(length(s(z0))) → c28(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(false)) → c28(LENGTH(ok(false)), PROPER(false))
PROPER(length(inf(z0))) → c28(LENGTH(inf(proper(z0))), PROPER(inf(z0)))
PROPER(length(cons(z0, z1))) → c28(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(take(z0, z1))) → c28(LENGTH(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(length(nil)) → c28(LENGTH(ok(nil)), PROPER(nil))
PROPER(length(length(z0))) → c28(LENGTH(length(proper(z0))), PROPER(length(z0)))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0))
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0))
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0))
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0))
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1))
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1))
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1))
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0)))
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(0)) → c22(S(ok(0)))
PROPER(s(true)) → c22(S(ok(true)))
PROPER(s(false)) → c22(S(ok(false)))
PROPER(s(nil)) → c22(S(ok(nil)))
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0)))
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0)))
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0)))
PROPER(inf(0)) → c24(INF(ok(0)))
PROPER(inf(true)) → c24(INF(ok(true)))
PROPER(inf(false)) → c24(INF(ok(false)))
PROPER(inf(nil)) → c24(INF(ok(nil)))
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0))
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0))
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1))
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0))
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0))
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0))
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0))
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(x1))
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(x1))
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(x1))
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(x1))
PROPER(length(eq(z0, z1))) → c28(LENGTH(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(length(0)) → c28(LENGTH(ok(0)), PROPER(0))
PROPER(length(true)) → c28(LENGTH(ok(true)), PROPER(true))
PROPER(length(s(z0))) → c28(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(false)) → c28(LENGTH(ok(false)), PROPER(false))
PROPER(length(inf(z0))) → c28(LENGTH(inf(proper(z0))), PROPER(inf(z0)))
PROPER(length(cons(z0, z1))) → c28(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(take(z0, z1))) → c28(LENGTH(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(length(nil)) → c28(LENGTH(ok(nil)), PROPER(nil))
PROPER(length(length(z0))) → c28(LENGTH(length(proper(z0))), PROPER(length(z0)))

K tuples:none
Defined Rule Symbols:

active, cons, take, s, length, inf, proper, eq

Defined Pair Symbols:

ACTIVE, INF, TAKE, LENGTH, EQ, S, CONS, TOP, PROPER

Compound Symbols:

c1, c9, c10, c12, c13, c14, c15, c16, c17, c18, c29, c30, c31, c32, c33, c3, c, c8, c8, c2, c11, c11, c4, c19, c19, c22, c22, c24, c24, c25, c25, c26, c26, c28

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

Removed 4 trailing tuple parts

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
s(ok(z0)) → ok(s(z0))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0))
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0))
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0))
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0))
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1))
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1))
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1))
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0)))
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(0)) → c22(S(ok(0)))
PROPER(s(true)) → c22(S(ok(true)))
PROPER(s(false)) → c22(S(ok(false)))
PROPER(s(nil)) → c22(S(ok(nil)))
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0)))
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0)))
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0)))
PROPER(inf(0)) → c24(INF(ok(0)))
PROPER(inf(true)) → c24(INF(ok(true)))
PROPER(inf(false)) → c24(INF(ok(false)))
PROPER(inf(nil)) → c24(INF(ok(nil)))
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0))
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0))
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1))
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0))
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0))
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0))
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0))
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(x1))
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(x1))
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(x1))
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(x1))
PROPER(length(eq(z0, z1))) → c28(LENGTH(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(length(s(z0))) → c28(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(inf(z0))) → c28(LENGTH(inf(proper(z0))), PROPER(inf(z0)))
PROPER(length(cons(z0, z1))) → c28(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(take(z0, z1))) → c28(LENGTH(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(length(length(z0))) → c28(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(0)) → c28(LENGTH(ok(0)))
PROPER(length(true)) → c28(LENGTH(ok(true)))
PROPER(length(false)) → c28(LENGTH(ok(false)))
PROPER(length(nil)) → c28(LENGTH(ok(nil)))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0))
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0))
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0))
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0))
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1))
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1))
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1))
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0)))
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(0)) → c22(S(ok(0)))
PROPER(s(true)) → c22(S(ok(true)))
PROPER(s(false)) → c22(S(ok(false)))
PROPER(s(nil)) → c22(S(ok(nil)))
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0)))
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0)))
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0)))
PROPER(inf(0)) → c24(INF(ok(0)))
PROPER(inf(true)) → c24(INF(ok(true)))
PROPER(inf(false)) → c24(INF(ok(false)))
PROPER(inf(nil)) → c24(INF(ok(nil)))
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0))
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0))
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1))
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0))
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0))
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0))
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0))
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(x1))
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(x1))
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(x1))
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(x1))
PROPER(length(eq(z0, z1))) → c28(LENGTH(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(length(s(z0))) → c28(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(inf(z0))) → c28(LENGTH(inf(proper(z0))), PROPER(inf(z0)))
PROPER(length(cons(z0, z1))) → c28(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(take(z0, z1))) → c28(LENGTH(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(length(length(z0))) → c28(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(0)) → c28(LENGTH(ok(0)))
PROPER(length(true)) → c28(LENGTH(ok(true)))
PROPER(length(false)) → c28(LENGTH(ok(false)))
PROPER(length(nil)) → c28(LENGTH(ok(nil)))

K tuples:none
Defined Rule Symbols:

active, cons, take, s, length, inf, proper, eq

Defined Pair Symbols:

ACTIVE, INF, TAKE, LENGTH, EQ, S, CONS, TOP, PROPER

Compound Symbols:

c1, c9, c10, c12, c13, c14, c15, c16, c17, c18, c29, c30, c31, c32, c33, c3, c, c8, c8, c2, c11, c11, c4, c19, c19, c22, c22, c24, c24, c25, c25, c26, c26, c28, c28

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

Use narrowing to replace TOP(mark(z0)) → c32(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
TOP(mark(0)) → c32(TOP(ok(0)), PROPER(0))
TOP(mark(true)) → c32(TOP(ok(true)), PROPER(true))
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(false)) → c32(TOP(ok(false)), PROPER(false))
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0)))
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
TOP(mark(nil)) → c32(TOP(ok(nil)), PROPER(nil))
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0)))

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
s(ok(z0)) → ok(s(z0))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0))
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0))
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0))
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0))
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1))
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1))
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1))
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0)))
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(0)) → c22(S(ok(0)))
PROPER(s(true)) → c22(S(ok(true)))
PROPER(s(false)) → c22(S(ok(false)))
PROPER(s(nil)) → c22(S(ok(nil)))
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0)))
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0)))
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0)))
PROPER(inf(0)) → c24(INF(ok(0)))
PROPER(inf(true)) → c24(INF(ok(true)))
PROPER(inf(false)) → c24(INF(ok(false)))
PROPER(inf(nil)) → c24(INF(ok(nil)))
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0))
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0))
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1))
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0))
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0))
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0))
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0))
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(x1))
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(x1))
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(x1))
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(x1))
PROPER(length(eq(z0, z1))) → c28(LENGTH(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(length(s(z0))) → c28(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(inf(z0))) → c28(LENGTH(inf(proper(z0))), PROPER(inf(z0)))
PROPER(length(cons(z0, z1))) → c28(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(take(z0, z1))) → c28(LENGTH(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(length(length(z0))) → c28(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(0)) → c28(LENGTH(ok(0)))
PROPER(length(true)) → c28(LENGTH(ok(true)))
PROPER(length(false)) → c28(LENGTH(ok(false)))
PROPER(length(nil)) → c28(LENGTH(ok(nil)))
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
TOP(mark(0)) → c32(TOP(ok(0)), PROPER(0))
TOP(mark(true)) → c32(TOP(ok(true)), PROPER(true))
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(false)) → c32(TOP(ok(false)), PROPER(false))
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0)))
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
TOP(mark(nil)) → c32(TOP(ok(nil)), PROPER(nil))
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0)))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0))
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0))
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0))
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0))
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1))
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1))
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1))
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0)))
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(0)) → c22(S(ok(0)))
PROPER(s(true)) → c22(S(ok(true)))
PROPER(s(false)) → c22(S(ok(false)))
PROPER(s(nil)) → c22(S(ok(nil)))
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0)))
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0)))
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0)))
PROPER(inf(0)) → c24(INF(ok(0)))
PROPER(inf(true)) → c24(INF(ok(true)))
PROPER(inf(false)) → c24(INF(ok(false)))
PROPER(inf(nil)) → c24(INF(ok(nil)))
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0))
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0))
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1))
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0))
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0))
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0))
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0))
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(x1))
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(x1))
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(x1))
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(x1))
PROPER(length(eq(z0, z1))) → c28(LENGTH(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(length(s(z0))) → c28(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(inf(z0))) → c28(LENGTH(inf(proper(z0))), PROPER(inf(z0)))
PROPER(length(cons(z0, z1))) → c28(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(take(z0, z1))) → c28(LENGTH(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(length(length(z0))) → c28(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(0)) → c28(LENGTH(ok(0)))
PROPER(length(true)) → c28(LENGTH(ok(true)))
PROPER(length(false)) → c28(LENGTH(ok(false)))
PROPER(length(nil)) → c28(LENGTH(ok(nil)))
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
TOP(mark(0)) → c32(TOP(ok(0)), PROPER(0))
TOP(mark(true)) → c32(TOP(ok(true)), PROPER(true))
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(false)) → c32(TOP(ok(false)), PROPER(false))
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0)))
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
TOP(mark(nil)) → c32(TOP(ok(nil)), PROPER(nil))
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0)))

K tuples:none
Defined Rule Symbols:

active, cons, take, s, length, inf, proper, eq

Defined Pair Symbols:

ACTIVE, INF, TAKE, LENGTH, EQ, S, CONS, TOP, PROPER

Compound Symbols:

c1, c9, c10, c12, c13, c14, c15, c16, c17, c18, c29, c30, c31, c33, c3, c, c8, c8, c2, c11, c11, c4, c19, c19, c22, c22, c24, c24, c25, c25, c26, c26, c28, c28, c32

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

Removed 4 trailing tuple parts

(50) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
s(ok(z0)) → ok(s(z0))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0))
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0))
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0))
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0))
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1))
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1))
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1))
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0)))
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(0)) → c22(S(ok(0)))
PROPER(s(true)) → c22(S(ok(true)))
PROPER(s(false)) → c22(S(ok(false)))
PROPER(s(nil)) → c22(S(ok(nil)))
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0)))
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0)))
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0)))
PROPER(inf(0)) → c24(INF(ok(0)))
PROPER(inf(true)) → c24(INF(ok(true)))
PROPER(inf(false)) → c24(INF(ok(false)))
PROPER(inf(nil)) → c24(INF(ok(nil)))
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0))
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0))
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1))
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0))
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0))
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0))
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0))
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(x1))
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(x1))
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(x1))
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(x1))
PROPER(length(eq(z0, z1))) → c28(LENGTH(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(length(s(z0))) → c28(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(inf(z0))) → c28(LENGTH(inf(proper(z0))), PROPER(inf(z0)))
PROPER(length(cons(z0, z1))) → c28(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(take(z0, z1))) → c28(LENGTH(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(length(length(z0))) → c28(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(0)) → c28(LENGTH(ok(0)))
PROPER(length(true)) → c28(LENGTH(ok(true)))
PROPER(length(false)) → c28(LENGTH(ok(false)))
PROPER(length(nil)) → c28(LENGTH(ok(nil)))
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0)))
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0)))
TOP(mark(0)) → c32(TOP(ok(0)))
TOP(mark(true)) → c32(TOP(ok(true)))
TOP(mark(false)) → c32(TOP(ok(false)))
TOP(mark(nil)) → c32(TOP(ok(nil)))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0))
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0))
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0))
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0))
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1))
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1))
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1))
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0)))
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(0)) → c22(S(ok(0)))
PROPER(s(true)) → c22(S(ok(true)))
PROPER(s(false)) → c22(S(ok(false)))
PROPER(s(nil)) → c22(S(ok(nil)))
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0)))
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0)))
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0)))
PROPER(inf(0)) → c24(INF(ok(0)))
PROPER(inf(true)) → c24(INF(ok(true)))
PROPER(inf(false)) → c24(INF(ok(false)))
PROPER(inf(nil)) → c24(INF(ok(nil)))
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0))
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0))
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1))
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0))
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0))
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0))
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0))
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(x1))
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(x1))
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(x1))
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(x1))
PROPER(length(eq(z0, z1))) → c28(LENGTH(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(length(s(z0))) → c28(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(inf(z0))) → c28(LENGTH(inf(proper(z0))), PROPER(inf(z0)))
PROPER(length(cons(z0, z1))) → c28(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(take(z0, z1))) → c28(LENGTH(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(length(length(z0))) → c28(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(0)) → c28(LENGTH(ok(0)))
PROPER(length(true)) → c28(LENGTH(ok(true)))
PROPER(length(false)) → c28(LENGTH(ok(false)))
PROPER(length(nil)) → c28(LENGTH(ok(nil)))
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0)))
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0)))
TOP(mark(0)) → c32(TOP(ok(0)))
TOP(mark(true)) → c32(TOP(ok(true)))
TOP(mark(false)) → c32(TOP(ok(false)))
TOP(mark(nil)) → c32(TOP(ok(nil)))

K tuples:none
Defined Rule Symbols:

active, cons, take, s, length, inf, proper, eq

Defined Pair Symbols:

ACTIVE, INF, TAKE, LENGTH, EQ, S, CONS, TOP, PROPER

Compound Symbols:

c1, c9, c10, c12, c13, c14, c15, c16, c17, c18, c29, c30, c31, c33, c3, c, c8, c8, c2, c11, c11, c4, c19, c19, c22, c22, c24, c24, c25, c25, c26, c26, c28, c28, c32, c32

(51) 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(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(0)) → c32(TOP(ok(0)))

We considered the (Usable) Rules:

active(length(nil)) → mark(0)
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
active(eq(z0, z1)) → mark(false)
take(mark(z0), z1) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
active(take(z0, z1)) → take(z0, active(z1))
length(mark(z0)) → mark(length(z0))
take(z0, mark(z1)) → mark(take(z0, z1))
active(inf(z0)) → inf(active(z0))
active(eq(0, 0)) → mark(true)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(z0, z1)) → take(active(z0), z1)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(z0)) → length(active(z0))
length(ok(z0)) → ok(length(z0))
active(length(cons(z0, z1))) → mark(s(length(z1)))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))
s(ok(z0)) → ok(s(z0))
active(take(0, z0)) → mark(nil)

And the Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0))
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0))
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0))
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0))
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1))
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1))
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1))
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0)))
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(0)) → c22(S(ok(0)))
PROPER(s(true)) → c22(S(ok(true)))
PROPER(s(false)) → c22(S(ok(false)))
PROPER(s(nil)) → c22(S(ok(nil)))
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0)))
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0)))
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0)))
PROPER(inf(0)) → c24(INF(ok(0)))
PROPER(inf(true)) → c24(INF(ok(true)))
PROPER(inf(false)) → c24(INF(ok(false)))
PROPER(inf(nil)) → c24(INF(ok(nil)))
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0))
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0))
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1))
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0))
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0))
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0))
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0))
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(x1))
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(x1))
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(x1))
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(x1))
PROPER(length(eq(z0, z1))) → c28(LENGTH(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(length(s(z0))) → c28(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(inf(z0))) → c28(LENGTH(inf(proper(z0))), PROPER(inf(z0)))
PROPER(length(cons(z0, z1))) → c28(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(take(z0, z1))) → c28(LENGTH(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(length(length(z0))) → c28(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(0)) → c28(LENGTH(ok(0)))
PROPER(length(true)) → c28(LENGTH(ok(true)))
PROPER(length(false)) → c28(LENGTH(ok(false)))
PROPER(length(nil)) → c28(LENGTH(ok(nil)))
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0)))
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0)))
TOP(mark(0)) → c32(TOP(ok(0)))
TOP(mark(true)) → c32(TOP(ok(true)))
TOP(mark(false)) → c32(TOP(ok(false)))
TOP(mark(nil)) → c32(TOP(ok(nil)))

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(EQ(x₁, x₂)) = 0
POL(INF(x₁)) = 0
POL(LENGTH(x₁)) = 0
POL(PROPER(x₁)) = 0
POL(S(x₁)) = 0
POL(TAKE(x₁, x₂)) = 0
POL(TOP(x₁)) = x₁
POL(active(x₁)) = x₁
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁, x₂)) = x₁ + x₂
POL(c11(x₁)) = x₁
POL(c11(x₁, x₂)) = x₁ + x₂
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁, x₂)) = x₁ + x₂
POL(c19(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c2(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c22(x₁, x₂)) = x₁ + x₂
POL(c24(x₁)) = x₁
POL(c24(x₁, x₂)) = x₁ + x₂
POL(c25(x₁, x₂)) = x₁ + x₂
POL(c25(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c26(x₁, x₂)) = x₁ + x₂
POL(c26(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c28(x₁)) = x₁
POL(c28(x₁, x₂)) = x₁ + x₂
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c32(x₁, x₂)) = x₁ + x₂
POL(c33(x₁, x₂)) = x₁ + x₂
POL(c4(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c8(x₁, x₂)) = x₁ + x₂
POL(c9(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = 0
POL(eq(x₁, x₂)) = [2]
POL(false) = [2]
POL(inf(x₁)) = [2]
POL(length(x₁)) = [2]
POL(mark(x₁)) = [2]
POL(nil) = [2]
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(s(x₁)) = [2]
POL(take(x₁, x₂)) = [2]
POL(true) = [2]

(52) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
s(ok(z0)) → ok(s(z0))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0))
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0))
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0))
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0))
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1))
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1))
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1))
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0)))
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(0)) → c22(S(ok(0)))
PROPER(s(true)) → c22(S(ok(true)))
PROPER(s(false)) → c22(S(ok(false)))
PROPER(s(nil)) → c22(S(ok(nil)))
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0)))
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0)))
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0)))
PROPER(inf(0)) → c24(INF(ok(0)))
PROPER(inf(true)) → c24(INF(ok(true)))
PROPER(inf(false)) → c24(INF(ok(false)))
PROPER(inf(nil)) → c24(INF(ok(nil)))
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0))
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0))
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1))
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0))
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0))
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0))
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0))
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(x1))
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(x1))
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(x1))
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(x1))
PROPER(length(eq(z0, z1))) → c28(LENGTH(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(length(s(z0))) → c28(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(inf(z0))) → c28(LENGTH(inf(proper(z0))), PROPER(inf(z0)))
PROPER(length(cons(z0, z1))) → c28(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(take(z0, z1))) → c28(LENGTH(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(length(length(z0))) → c28(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(0)) → c28(LENGTH(ok(0)))
PROPER(length(true)) → c28(LENGTH(ok(true)))
PROPER(length(false)) → c28(LENGTH(ok(false)))
PROPER(length(nil)) → c28(LENGTH(ok(nil)))
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0)))
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0)))
TOP(mark(0)) → c32(TOP(ok(0)))
TOP(mark(true)) → c32(TOP(ok(true)))
TOP(mark(false)) → c32(TOP(ok(false)))
TOP(mark(nil)) → c32(TOP(ok(nil)))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0))
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0))
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0))
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0))
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1))
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1))
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1))
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0)))
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(0)) → c22(S(ok(0)))
PROPER(s(true)) → c22(S(ok(true)))
PROPER(s(false)) → c22(S(ok(false)))
PROPER(s(nil)) → c22(S(ok(nil)))
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0)))
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0)))
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0)))
PROPER(inf(0)) → c24(INF(ok(0)))
PROPER(inf(true)) → c24(INF(ok(true)))
PROPER(inf(false)) → c24(INF(ok(false)))
PROPER(inf(nil)) → c24(INF(ok(nil)))
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0))
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0))
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1))
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0))
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0))
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0))
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0))
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(x1))
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(x1))
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(x1))
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(x1))
PROPER(length(eq(z0, z1))) → c28(LENGTH(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(length(s(z0))) → c28(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(inf(z0))) → c28(LENGTH(inf(proper(z0))), PROPER(inf(z0)))
PROPER(length(cons(z0, z1))) → c28(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(take(z0, z1))) → c28(LENGTH(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(length(length(z0))) → c28(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(0)) → c28(LENGTH(ok(0)))
PROPER(length(true)) → c28(LENGTH(ok(true)))
PROPER(length(false)) → c28(LENGTH(ok(false)))
PROPER(length(nil)) → c28(LENGTH(ok(nil)))
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0)))
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0)))
TOP(mark(true)) → c32(TOP(ok(true)))
TOP(mark(false)) → c32(TOP(ok(false)))
TOP(mark(nil)) → c32(TOP(ok(nil)))

K tuples:

TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(0)) → c32(TOP(ok(0)))

Defined Rule Symbols:

active, cons, take, s, length, inf, proper, eq

Defined Pair Symbols:

ACTIVE, INF, TAKE, LENGTH, EQ, S, CONS, TOP, PROPER

Compound Symbols:

c1, c9, c10, c12, c13, c14, c15, c16, c17, c18, c29, c30, c31, c33, c3, c, c8, c8, c2, c11, c11, c4, c19, c19, c22, c22, c24, c24, c25, c25, c26, c26, c28, c28, c32, c32

(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(true)) → c32(TOP(ok(true)))
TOP(mark(false)) → c32(TOP(ok(false)))
TOP(mark(nil)) → c32(TOP(ok(nil)))

We considered the (Usable) Rules:

active(length(nil)) → mark(0)
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
active(eq(z0, z1)) → mark(false)
take(mark(z0), z1) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
active(take(z0, z1)) → take(z0, active(z1))
length(mark(z0)) → mark(length(z0))
take(z0, mark(z1)) → mark(take(z0, z1))
active(inf(z0)) → inf(active(z0))
active(eq(0, 0)) → mark(true)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(z0, z1)) → take(active(z0), z1)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(z0)) → length(active(z0))
length(ok(z0)) → ok(length(z0))
active(length(cons(z0, z1))) → mark(s(length(z1)))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))
s(ok(z0)) → ok(s(z0))
active(take(0, z0)) → mark(nil)

And the Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0))
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0))
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0))
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0))
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1))
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1))
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1))
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0)))
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(0)) → c22(S(ok(0)))
PROPER(s(true)) → c22(S(ok(true)))
PROPER(s(false)) → c22(S(ok(false)))
PROPER(s(nil)) → c22(S(ok(nil)))
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0)))
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0)))
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0)))
PROPER(inf(0)) → c24(INF(ok(0)))
PROPER(inf(true)) → c24(INF(ok(true)))
PROPER(inf(false)) → c24(INF(ok(false)))
PROPER(inf(nil)) → c24(INF(ok(nil)))
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0))
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0))
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1))
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0))
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0))
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0))
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0))
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(x1))
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(x1))
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(x1))
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(x1))
PROPER(length(eq(z0, z1))) → c28(LENGTH(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(length(s(z0))) → c28(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(inf(z0))) → c28(LENGTH(inf(proper(z0))), PROPER(inf(z0)))
PROPER(length(cons(z0, z1))) → c28(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(take(z0, z1))) → c28(LENGTH(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(length(length(z0))) → c28(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(0)) → c28(LENGTH(ok(0)))
PROPER(length(true)) → c28(LENGTH(ok(true)))
PROPER(length(false)) → c28(LENGTH(ok(false)))
PROPER(length(nil)) → c28(LENGTH(ok(nil)))
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0)))
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0)))
TOP(mark(0)) → c32(TOP(ok(0)))
TOP(mark(true)) → c32(TOP(ok(true)))
TOP(mark(false)) → c32(TOP(ok(false)))
TOP(mark(nil)) → c32(TOP(ok(nil)))

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

POL(0) = [4]
POL(ACTIVE(x₁)) = 0
POL(CONS(x₁, x₂)) = 0
POL(EQ(x₁, x₂)) = 0
POL(INF(x₁)) = 0
POL(LENGTH(x₁)) = 0
POL(PROPER(x₁)) = 0
POL(S(x₁)) = 0
POL(TAKE(x₁, x₂)) = 0
POL(TOP(x₁)) = [2]x₁
POL(active(x₁)) = x₁
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁, x₂)) = x₁ + x₂
POL(c11(x₁)) = x₁
POL(c11(x₁, x₂)) = x₁ + x₂
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁, x₂)) = x₁ + x₂
POL(c19(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c2(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c22(x₁, x₂)) = x₁ + x₂
POL(c24(x₁)) = x₁
POL(c24(x₁, x₂)) = x₁ + x₂
POL(c25(x₁, x₂)) = x₁ + x₂
POL(c25(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c26(x₁, x₂)) = x₁ + x₂
POL(c26(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c28(x₁)) = x₁
POL(c28(x₁, x₂)) = x₁ + x₂
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c32(x₁, x₂)) = x₁ + x₂
POL(c33(x₁, x₂)) = x₁ + x₂
POL(c4(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c8(x₁, x₂)) = x₁ + x₂
POL(c9(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = [1]
POL(eq(x₁, x₂)) = [4]
POL(false) = 0
POL(inf(x₁)) = [4]
POL(length(x₁)) = [4]
POL(mark(x₁)) = [4]
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(s(x₁)) = [4]
POL(take(x₁, x₂)) = [4]
POL(true) = [1]

(54) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
s(ok(z0)) → ok(s(z0))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0))
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0))
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0))
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0))
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1))
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1))
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1))
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0)))
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(0)) → c22(S(ok(0)))
PROPER(s(true)) → c22(S(ok(true)))
PROPER(s(false)) → c22(S(ok(false)))
PROPER(s(nil)) → c22(S(ok(nil)))
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0)))
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0)))
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0)))
PROPER(inf(0)) → c24(INF(ok(0)))
PROPER(inf(true)) → c24(INF(ok(true)))
PROPER(inf(false)) → c24(INF(ok(false)))
PROPER(inf(nil)) → c24(INF(ok(nil)))
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0))
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0))
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1))
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0))
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0))
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0))
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0))
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(x1))
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(x1))
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(x1))
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(x1))
PROPER(length(eq(z0, z1))) → c28(LENGTH(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(length(s(z0))) → c28(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(inf(z0))) → c28(LENGTH(inf(proper(z0))), PROPER(inf(z0)))
PROPER(length(cons(z0, z1))) → c28(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(take(z0, z1))) → c28(LENGTH(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(length(length(z0))) → c28(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(0)) → c28(LENGTH(ok(0)))
PROPER(length(true)) → c28(LENGTH(ok(true)))
PROPER(length(false)) → c28(LENGTH(ok(false)))
PROPER(length(nil)) → c28(LENGTH(ok(nil)))
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0)))
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0)))
TOP(mark(0)) → c32(TOP(ok(0)))
TOP(mark(true)) → c32(TOP(ok(true)))
TOP(mark(false)) → c32(TOP(ok(false)))
TOP(mark(nil)) → c32(TOP(ok(nil)))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0))
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0))
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0))
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0))
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1))
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1))
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1))
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0)))
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(0)) → c22(S(ok(0)))
PROPER(s(true)) → c22(S(ok(true)))
PROPER(s(false)) → c22(S(ok(false)))
PROPER(s(nil)) → c22(S(ok(nil)))
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0)))
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0)))
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0)))
PROPER(inf(0)) → c24(INF(ok(0)))
PROPER(inf(true)) → c24(INF(ok(true)))
PROPER(inf(false)) → c24(INF(ok(false)))
PROPER(inf(nil)) → c24(INF(ok(nil)))
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0))
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0))
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1))
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0))
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0))
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0))
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0))
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(x1))
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(x1))
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(x1))
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(x1))
PROPER(length(eq(z0, z1))) → c28(LENGTH(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(length(s(z0))) → c28(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(inf(z0))) → c28(LENGTH(inf(proper(z0))), PROPER(inf(z0)))
PROPER(length(cons(z0, z1))) → c28(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(take(z0, z1))) → c28(LENGTH(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(length(length(z0))) → c28(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(0)) → c28(LENGTH(ok(0)))
PROPER(length(true)) → c28(LENGTH(ok(true)))
PROPER(length(false)) → c28(LENGTH(ok(false)))
PROPER(length(nil)) → c28(LENGTH(ok(nil)))
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0)))
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0)))

K tuples:

TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(0)) → c32(TOP(ok(0)))
TOP(mark(true)) → c32(TOP(ok(true)))
TOP(mark(false)) → c32(TOP(ok(false)))
TOP(mark(nil)) → c32(TOP(ok(nil)))

Defined Rule Symbols:

active, cons, take, s, length, inf, proper, eq

Defined Pair Symbols:

ACTIVE, INF, TAKE, LENGTH, EQ, S, CONS, TOP, PROPER

Compound Symbols:

c1, c9, c10, c12, c13, c14, c15, c16, c17, c18, c29, c30, c31, c33, c3, c, c8, c8, c2, c11, c11, c4, c19, c19, c22, c22, c24, c24, c25, c25, c26, c26, c28, c28, c32, c32

(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(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0)))

We considered the (Usable) Rules:

active(length(nil)) → mark(0)
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
active(eq(z0, z1)) → mark(false)
take(mark(z0), z1) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
active(take(z0, z1)) → take(z0, active(z1))
length(mark(z0)) → mark(length(z0))
take(z0, mark(z1)) → mark(take(z0, z1))
active(inf(z0)) → inf(active(z0))
active(eq(0, 0)) → mark(true)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(z0, z1)) → take(active(z0), z1)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(z0)) → length(active(z0))
length(ok(z0)) → ok(length(z0))
active(length(cons(z0, z1))) → mark(s(length(z1)))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))
s(ok(z0)) → ok(s(z0))
active(take(0, z0)) → mark(nil)

And the Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0))
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0))
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0))
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0))
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1))
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1))
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1))
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0)))
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(0)) → c22(S(ok(0)))
PROPER(s(true)) → c22(S(ok(true)))
PROPER(s(false)) → c22(S(ok(false)))
PROPER(s(nil)) → c22(S(ok(nil)))
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0)))
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0)))
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0)))
PROPER(inf(0)) → c24(INF(ok(0)))
PROPER(inf(true)) → c24(INF(ok(true)))
PROPER(inf(false)) → c24(INF(ok(false)))
PROPER(inf(nil)) → c24(INF(ok(nil)))
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0))
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0))
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1))
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0))
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0))
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0))
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0))
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(x1))
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(x1))
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(x1))
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(x1))
PROPER(length(eq(z0, z1))) → c28(LENGTH(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(length(s(z0))) → c28(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(inf(z0))) → c28(LENGTH(inf(proper(z0))), PROPER(inf(z0)))
PROPER(length(cons(z0, z1))) → c28(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(take(z0, z1))) → c28(LENGTH(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(length(length(z0))) → c28(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(0)) → c28(LENGTH(ok(0)))
PROPER(length(true)) → c28(LENGTH(ok(true)))
PROPER(length(false)) → c28(LENGTH(ok(false)))
PROPER(length(nil)) → c28(LENGTH(ok(nil)))
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0)))
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0)))
TOP(mark(0)) → c32(TOP(ok(0)))
TOP(mark(true)) → c32(TOP(ok(true)))
TOP(mark(false)) → c32(TOP(ok(false)))
TOP(mark(nil)) → c32(TOP(ok(nil)))

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(EQ(x₁, x₂)) = 0
POL(INF(x₁)) = 0
POL(LENGTH(x₁)) = 0
POL(PROPER(x₁)) = 0
POL(S(x₁)) = 0
POL(TAKE(x₁, x₂)) = 0
POL(TOP(x₁)) = x₁
POL(active(x₁)) = x₁
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁, x₂)) = x₁ + x₂
POL(c11(x₁)) = x₁
POL(c11(x₁, x₂)) = x₁ + x₂
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁, x₂)) = x₁ + x₂
POL(c19(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c2(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c22(x₁, x₂)) = x₁ + x₂
POL(c24(x₁)) = x₁
POL(c24(x₁, x₂)) = x₁ + x₂
POL(c25(x₁, x₂)) = x₁ + x₂
POL(c25(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c26(x₁, x₂)) = x₁ + x₂
POL(c26(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c28(x₁)) = x₁
POL(c28(x₁, x₂)) = x₁ + x₂
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c32(x₁, x₂)) = x₁ + x₂
POL(c33(x₁, x₂)) = x₁ + x₂
POL(c4(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c8(x₁, x₂)) = x₁ + x₂
POL(c9(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = [2]
POL(eq(x₁, x₂)) = [2]
POL(false) = 0
POL(inf(x₁)) = [2]
POL(length(x₁)) = [2]
POL(mark(x₁)) = [2]
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = [2]
POL(true) = 0

(56) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
s(ok(z0)) → ok(s(z0))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0))
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0))
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0))
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0))
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1))
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1))
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1))
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0)))
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(0)) → c22(S(ok(0)))
PROPER(s(true)) → c22(S(ok(true)))
PROPER(s(false)) → c22(S(ok(false)))
PROPER(s(nil)) → c22(S(ok(nil)))
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0)))
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0)))
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0)))
PROPER(inf(0)) → c24(INF(ok(0)))
PROPER(inf(true)) → c24(INF(ok(true)))
PROPER(inf(false)) → c24(INF(ok(false)))
PROPER(inf(nil)) → c24(INF(ok(nil)))
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0))
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0))
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1))
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0))
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0))
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0))
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0))
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(x1))
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(x1))
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(x1))
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(x1))
PROPER(length(eq(z0, z1))) → c28(LENGTH(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(length(s(z0))) → c28(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(inf(z0))) → c28(LENGTH(inf(proper(z0))), PROPER(inf(z0)))
PROPER(length(cons(z0, z1))) → c28(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(take(z0, z1))) → c28(LENGTH(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(length(length(z0))) → c28(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(0)) → c28(LENGTH(ok(0)))
PROPER(length(true)) → c28(LENGTH(ok(true)))
PROPER(length(false)) → c28(LENGTH(ok(false)))
PROPER(length(nil)) → c28(LENGTH(ok(nil)))
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0)))
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0)))
TOP(mark(0)) → c32(TOP(ok(0)))
TOP(mark(true)) → c32(TOP(ok(true)))
TOP(mark(false)) → c32(TOP(ok(false)))
TOP(mark(nil)) → c32(TOP(ok(nil)))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0))
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0))
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0))
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0))
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1))
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1))
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1))
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0)))
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(0)) → c22(S(ok(0)))
PROPER(s(true)) → c22(S(ok(true)))
PROPER(s(false)) → c22(S(ok(false)))
PROPER(s(nil)) → c22(S(ok(nil)))
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0)))
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0)))
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0)))
PROPER(inf(0)) → c24(INF(ok(0)))
PROPER(inf(true)) → c24(INF(ok(true)))
PROPER(inf(false)) → c24(INF(ok(false)))
PROPER(inf(nil)) → c24(INF(ok(nil)))
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0))
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0))
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1))
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0))
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0))
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0))
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0))
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(x1))
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(x1))
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(x1))
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(x1))
PROPER(length(eq(z0, z1))) → c28(LENGTH(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(length(s(z0))) → c28(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(inf(z0))) → c28(LENGTH(inf(proper(z0))), PROPER(inf(z0)))
PROPER(length(cons(z0, z1))) → c28(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(take(z0, z1))) → c28(LENGTH(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(length(length(z0))) → c28(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(0)) → c28(LENGTH(ok(0)))
PROPER(length(true)) → c28(LENGTH(ok(true)))
PROPER(length(false)) → c28(LENGTH(ok(false)))
PROPER(length(nil)) → c28(LENGTH(ok(nil)))
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0)))
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0)))

K tuples:

TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(0)) → c32(TOP(ok(0)))
TOP(mark(true)) → c32(TOP(ok(true)))
TOP(mark(false)) → c32(TOP(ok(false)))
TOP(mark(nil)) → c32(TOP(ok(nil)))
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0)))

Defined Rule Symbols:

active, cons, take, s, length, inf, proper, eq

Defined Pair Symbols:

ACTIVE, INF, TAKE, LENGTH, EQ, S, CONS, TOP, PROPER

Compound Symbols:

c1, c9, c10, c12, c13, c14, c15, c16, c17, c18, c29, c30, c31, c33, c3, c, c8, c8, c2, c11, c11, c4, c19, c19, c22, c22, c24, c24, c25, c25, c26, c26, c28, c28, c32, c32

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

Use narrowing to replace TOP(ok(z0)) → c33(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(eq(0, 0))) → c33(TOP(mark(true)), ACTIVE(eq(0, 0)))
TOP(ok(eq(s(z0), s(z1)))) → c33(TOP(mark(eq(z0, z1))), ACTIVE(eq(s(z0), s(z1))))
TOP(ok(eq(z0, z1))) → c33(TOP(mark(false)), ACTIVE(eq(z0, z1)))
TOP(ok(inf(z0))) → c33(TOP(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
TOP(ok(take(0, z0))) → c33(TOP(mark(nil)), ACTIVE(take(0, z0)))
TOP(ok(take(s(z0), cons(z1, z2)))) → c33(TOP(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
TOP(ok(length(nil))) → c33(TOP(mark(0)), ACTIVE(length(nil)))
TOP(ok(length(cons(z0, z1)))) → c33(TOP(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
TOP(ok(inf(z0))) → c33(TOP(inf(active(z0))), ACTIVE(inf(z0)))
TOP(ok(take(z0, z1))) → c33(TOP(take(active(z0), z1)), ACTIVE(take(z0, z1)))
TOP(ok(take(z0, z1))) → c33(TOP(take(z0, active(z1))), ACTIVE(take(z0, z1)))
TOP(ok(length(z0))) → c33(TOP(length(active(z0))), ACTIVE(length(z0)))

(58) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
s(ok(z0)) → ok(s(z0))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0))
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0))
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0))
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0))
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1))
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1))
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1))
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0)))
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(0)) → c22(S(ok(0)))
PROPER(s(true)) → c22(S(ok(true)))
PROPER(s(false)) → c22(S(ok(false)))
PROPER(s(nil)) → c22(S(ok(nil)))
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0)))
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0)))
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0)))
PROPER(inf(0)) → c24(INF(ok(0)))
PROPER(inf(true)) → c24(INF(ok(true)))
PROPER(inf(false)) → c24(INF(ok(false)))
PROPER(inf(nil)) → c24(INF(ok(nil)))
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0))
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0))
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1))
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0))
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0))
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0))
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0))
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(x1))
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(x1))
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(x1))
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(x1))
PROPER(length(eq(z0, z1))) → c28(LENGTH(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(length(s(z0))) → c28(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(inf(z0))) → c28(LENGTH(inf(proper(z0))), PROPER(inf(z0)))
PROPER(length(cons(z0, z1))) → c28(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(take(z0, z1))) → c28(LENGTH(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(length(length(z0))) → c28(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(0)) → c28(LENGTH(ok(0)))
PROPER(length(true)) → c28(LENGTH(ok(true)))
PROPER(length(false)) → c28(LENGTH(ok(false)))
PROPER(length(nil)) → c28(LENGTH(ok(nil)))
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0)))
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0)))
TOP(mark(0)) → c32(TOP(ok(0)))
TOP(mark(true)) → c32(TOP(ok(true)))
TOP(mark(false)) → c32(TOP(ok(false)))
TOP(mark(nil)) → c32(TOP(ok(nil)))
TOP(ok(eq(0, 0))) → c33(TOP(mark(true)), ACTIVE(eq(0, 0)))
TOP(ok(eq(s(z0), s(z1)))) → c33(TOP(mark(eq(z0, z1))), ACTIVE(eq(s(z0), s(z1))))
TOP(ok(eq(z0, z1))) → c33(TOP(mark(false)), ACTIVE(eq(z0, z1)))
TOP(ok(inf(z0))) → c33(TOP(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
TOP(ok(take(0, z0))) → c33(TOP(mark(nil)), ACTIVE(take(0, z0)))
TOP(ok(take(s(z0), cons(z1, z2)))) → c33(TOP(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
TOP(ok(length(nil))) → c33(TOP(mark(0)), ACTIVE(length(nil)))
TOP(ok(length(cons(z0, z1)))) → c33(TOP(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
TOP(ok(inf(z0))) → c33(TOP(inf(active(z0))), ACTIVE(inf(z0)))
TOP(ok(take(z0, z1))) → c33(TOP(take(active(z0), z1)), ACTIVE(take(z0, z1)))
TOP(ok(take(z0, z1))) → c33(TOP(take(z0, active(z1))), ACTIVE(take(z0, z1)))
TOP(ok(length(z0))) → c33(TOP(length(active(z0))), ACTIVE(length(z0)))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0))
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0))
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0))
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0))
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1))
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1))
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1))
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0)))
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(0)) → c22(S(ok(0)))
PROPER(s(true)) → c22(S(ok(true)))
PROPER(s(false)) → c22(S(ok(false)))
PROPER(s(nil)) → c22(S(ok(nil)))
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0)))
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0)))
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0)))
PROPER(inf(0)) → c24(INF(ok(0)))
PROPER(inf(true)) → c24(INF(ok(true)))
PROPER(inf(false)) → c24(INF(ok(false)))
PROPER(inf(nil)) → c24(INF(ok(nil)))
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0))
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0))
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1))
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0))
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0))
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0))
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0))
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(x1))
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(x1))
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(x1))
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(x1))
PROPER(length(eq(z0, z1))) → c28(LENGTH(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(length(s(z0))) → c28(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(inf(z0))) → c28(LENGTH(inf(proper(z0))), PROPER(inf(z0)))
PROPER(length(cons(z0, z1))) → c28(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(take(z0, z1))) → c28(LENGTH(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(length(length(z0))) → c28(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(0)) → c28(LENGTH(ok(0)))
PROPER(length(true)) → c28(LENGTH(ok(true)))
PROPER(length(false)) → c28(LENGTH(ok(false)))
PROPER(length(nil)) → c28(LENGTH(ok(nil)))
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0)))
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0)))
TOP(ok(eq(0, 0))) → c33(TOP(mark(true)), ACTIVE(eq(0, 0)))
TOP(ok(eq(s(z0), s(z1)))) → c33(TOP(mark(eq(z0, z1))), ACTIVE(eq(s(z0), s(z1))))
TOP(ok(eq(z0, z1))) → c33(TOP(mark(false)), ACTIVE(eq(z0, z1)))
TOP(ok(inf(z0))) → c33(TOP(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
TOP(ok(take(0, z0))) → c33(TOP(mark(nil)), ACTIVE(take(0, z0)))
TOP(ok(take(s(z0), cons(z1, z2)))) → c33(TOP(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
TOP(ok(length(nil))) → c33(TOP(mark(0)), ACTIVE(length(nil)))
TOP(ok(length(cons(z0, z1)))) → c33(TOP(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
TOP(ok(inf(z0))) → c33(TOP(inf(active(z0))), ACTIVE(inf(z0)))
TOP(ok(take(z0, z1))) → c33(TOP(take(active(z0), z1)), ACTIVE(take(z0, z1)))
TOP(ok(take(z0, z1))) → c33(TOP(take(z0, active(z1))), ACTIVE(take(z0, z1)))
TOP(ok(length(z0))) → c33(TOP(length(active(z0))), ACTIVE(length(z0)))

K tuples:

TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(0)) → c32(TOP(ok(0)))
TOP(mark(true)) → c32(TOP(ok(true)))
TOP(mark(false)) → c32(TOP(ok(false)))
TOP(mark(nil)) → c32(TOP(ok(nil)))
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0)))

Defined Rule Symbols:

active, cons, take, s, length, inf, proper, eq

Defined Pair Symbols:

ACTIVE, INF, TAKE, LENGTH, EQ, S, CONS, PROPER, TOP

Compound Symbols:

c1, c9, c10, c12, c13, c14, c15, c16, c17, c18, c29, c30, c31, c3, c, c8, c8, c2, c11, c11, c4, c19, c19, c22, c22, c24, c24, c25, c25, c26, c26, c28, c28, c32, c32, c33

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

Removed 6 trailing nodes:

TOP(mark(true)) → c32(TOP(ok(true)))
TOP(mark(false)) → c32(TOP(ok(false)))
TOP(mark(nil)) → c32(TOP(ok(nil)))
TOP(ok(length(nil))) → c33(TOP(mark(0)), ACTIVE(length(nil)))
TOP(ok(eq(0, 0))) → c33(TOP(mark(true)), ACTIVE(eq(0, 0)))
TOP(mark(0)) → c32(TOP(ok(0)))

(60) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
s(ok(z0)) → ok(s(z0))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))

Tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0))
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0))
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0))
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0))
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1))
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1))
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1))
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0)))
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(0)) → c22(S(ok(0)))
PROPER(s(true)) → c22(S(ok(true)))
PROPER(s(false)) → c22(S(ok(false)))
PROPER(s(nil)) → c22(S(ok(nil)))
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0)))
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0)))
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0)))
PROPER(inf(0)) → c24(INF(ok(0)))
PROPER(inf(true)) → c24(INF(ok(true)))
PROPER(inf(false)) → c24(INF(ok(false)))
PROPER(inf(nil)) → c24(INF(ok(nil)))
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0))
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0))
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1))
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0))
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0))
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0))
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0))
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(x1))
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(x1))
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(x1))
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(x1))
PROPER(length(eq(z0, z1))) → c28(LENGTH(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(length(s(z0))) → c28(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(inf(z0))) → c28(LENGTH(inf(proper(z0))), PROPER(inf(z0)))
PROPER(length(cons(z0, z1))) → c28(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(take(z0, z1))) → c28(LENGTH(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(length(length(z0))) → c28(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(0)) → c28(LENGTH(ok(0)))
PROPER(length(true)) → c28(LENGTH(ok(true)))
PROPER(length(false)) → c28(LENGTH(ok(false)))
PROPER(length(nil)) → c28(LENGTH(ok(nil)))
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0)))
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0)))
TOP(ok(eq(s(z0), s(z1)))) → c33(TOP(mark(eq(z0, z1))), ACTIVE(eq(s(z0), s(z1))))
TOP(ok(eq(z0, z1))) → c33(TOP(mark(false)), ACTIVE(eq(z0, z1)))
TOP(ok(inf(z0))) → c33(TOP(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
TOP(ok(take(0, z0))) → c33(TOP(mark(nil)), ACTIVE(take(0, z0)))
TOP(ok(take(s(z0), cons(z1, z2)))) → c33(TOP(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
TOP(ok(length(cons(z0, z1)))) → c33(TOP(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
TOP(ok(inf(z0))) → c33(TOP(inf(active(z0))), ACTIVE(inf(z0)))
TOP(ok(take(z0, z1))) → c33(TOP(take(active(z0), z1)), ACTIVE(take(z0, z1)))
TOP(ok(take(z0, z1))) → c33(TOP(take(z0, active(z1))), ACTIVE(take(z0, z1)))
TOP(ok(length(z0))) → c33(TOP(length(active(z0))), ACTIVE(length(z0)))

S tuples:

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0))
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0))
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0))
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0))
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1))
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1))
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1))
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0)))
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(0)) → c22(S(ok(0)))
PROPER(s(true)) → c22(S(ok(true)))
PROPER(s(false)) → c22(S(ok(false)))
PROPER(s(nil)) → c22(S(ok(nil)))
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0)))
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0)))
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0)))
PROPER(inf(0)) → c24(INF(ok(0)))
PROPER(inf(true)) → c24(INF(ok(true)))
PROPER(inf(false)) → c24(INF(ok(false)))
PROPER(inf(nil)) → c24(INF(ok(nil)))
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0))
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0))
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1))
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0))
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0))
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0))
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0))
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(x1))
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(x1))
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(x1))
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(x1))
PROPER(length(eq(z0, z1))) → c28(LENGTH(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(length(s(z0))) → c28(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(inf(z0))) → c28(LENGTH(inf(proper(z0))), PROPER(inf(z0)))
PROPER(length(cons(z0, z1))) → c28(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(take(z0, z1))) → c28(LENGTH(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(length(length(z0))) → c28(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(0)) → c28(LENGTH(ok(0)))
PROPER(length(true)) → c28(LENGTH(ok(true)))
PROPER(length(false)) → c28(LENGTH(ok(false)))
PROPER(length(nil)) → c28(LENGTH(ok(nil)))
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0)))
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0)))
TOP(ok(eq(s(z0), s(z1)))) → c33(TOP(mark(eq(z0, z1))), ACTIVE(eq(s(z0), s(z1))))
TOP(ok(eq(z0, z1))) → c33(TOP(mark(false)), ACTIVE(eq(z0, z1)))
TOP(ok(inf(z0))) → c33(TOP(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
TOP(ok(take(0, z0))) → c33(TOP(mark(nil)), ACTIVE(take(0, z0)))
TOP(ok(take(s(z0), cons(z1, z2)))) → c33(TOP(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
TOP(ok(length(cons(z0, z1)))) → c33(TOP(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
TOP(ok(inf(z0))) → c33(TOP(inf(active(z0))), ACTIVE(inf(z0)))
TOP(ok(take(z0, z1))) → c33(TOP(take(active(z0), z1)), ACTIVE(take(z0, z1)))
TOP(ok(take(z0, z1))) → c33(TOP(take(z0, active(z1))), ACTIVE(take(z0, z1)))
TOP(ok(length(z0))) → c33(TOP(length(active(z0))), ACTIVE(length(z0)))

K tuples:

TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0)))

Defined Rule Symbols:

active, cons, take, s, length, inf, proper, eq

Defined Pair Symbols:

ACTIVE, INF, TAKE, LENGTH, EQ, S, CONS, PROPER, TOP

Compound Symbols:

c1, c9, c10, c12, c13, c14, c15, c16, c17, c18, c29, c30, c31, c3, c, c8, c8, c2, c11, c11, c4, c19, c19, c22, c22, c24, c24, c25, c25, c26, c26, c28, c28, c32, c33

(61) CdtUnreachableProof (EQUIVALENT transformation)

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

ACTIVE(eq(s(z0), s(z1))) → c1(EQ(z0, z1))
ACTIVE(take(z0, z1)) → c9(TAKE(active(z0), z1), ACTIVE(z0))
ACTIVE(take(z0, z1)) → c10(TAKE(z0, active(z1)), ACTIVE(z1))
ACTIVE(inf(z0)) → c3(S(z0))
ACTIVE(take(s(z0), cons(z1, z2))) → c(CONS(z1, take(z0, z2)))
ACTIVE(take(s(z0), cons(z1, z2))) → c(TAKE(z0, z2))
ACTIVE(length(cons(z0, z1))) → c(S(length(z1)))
ACTIVE(length(cons(z0, z1))) → c(LENGTH(z1))
ACTIVE(inf(inf(z0))) → c8(INF(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(inf(take(0, z0))) → c8(INF(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(inf(take(s(z0), cons(z1, z2)))) → c8(INF(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(inf(inf(z0))) → c8(INF(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(inf(take(z0, z1))) → c8(INF(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(inf(length(z0))) → c8(INF(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(eq(0, 0))) → c8(INF(mark(true)))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(INF(mark(eq(z0, z1))))
ACTIVE(inf(eq(s(z0), s(z1)))) → c2(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(inf(eq(z0, z1))) → c2(INF(mark(false)))
ACTIVE(inf(eq(z0, z1))) → c2(ACTIVE(eq(z0, z1)))
ACTIVE(length(inf(z0))) → c11(LENGTH(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
ACTIVE(length(take(0, z0))) → c11(LENGTH(mark(nil)), ACTIVE(take(0, z0)))
ACTIVE(length(take(s(z0), cons(z1, z2)))) → c11(LENGTH(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
ACTIVE(length(inf(z0))) → c11(LENGTH(inf(active(z0))), ACTIVE(inf(z0)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(active(z0), z1)), ACTIVE(take(z0, z1)))
ACTIVE(length(take(z0, z1))) → c11(LENGTH(take(z0, active(z1))), ACTIVE(take(z0, z1)))
ACTIVE(length(length(z0))) → c11(LENGTH(length(active(z0))), ACTIVE(length(z0)))
ACTIVE(inf(length(nil))) → c8(INF(mark(0)))
ACTIVE(length(eq(0, 0))) → c11(LENGTH(mark(true)))
ACTIVE(length(length(nil))) → c11(LENGTH(mark(0)))
ACTIVE(inf(length(cons(z0, z1)))) → c4(INF(mark(s(length(z1)))))
ACTIVE(inf(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(LENGTH(mark(eq(z0, z1))))
ACTIVE(length(eq(s(z0), s(z1)))) → c4(ACTIVE(eq(s(z0), s(z1))))
ACTIVE(length(eq(z0, z1))) → c4(LENGTH(mark(false)))
ACTIVE(length(eq(z0, z1))) → c4(ACTIVE(eq(z0, z1)))
ACTIVE(length(length(cons(z0, z1)))) → c4(LENGTH(mark(s(length(z1)))))
ACTIVE(length(length(cons(z0, z1)))) → c4(ACTIVE(length(cons(z0, z1))))
PROPER(eq(x0, eq(z0, z1))) → c19(EQ(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(eq(x0, s(z0))) → c19(EQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(eq(x0, inf(z0))) → c19(EQ(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(eq(x0, cons(z0, z1))) → c19(EQ(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(eq(x0, take(z0, z1))) → c19(EQ(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(eq(x0, length(z0))) → c19(EQ(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(eq(eq(z0, z1), x1)) → c19(EQ(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(eq(s(z0), x1)) → c19(EQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(eq(inf(z0), x1)) → c19(EQ(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(eq(cons(z0, z1), x1)) → c19(EQ(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(eq(take(z0, z1), x1)) → c19(EQ(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(eq(length(z0), x1)) → c19(EQ(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(eq(x0, 0)) → c19(EQ(proper(x0), ok(0)), PROPER(x0))
PROPER(eq(x0, true)) → c19(EQ(proper(x0), ok(true)), PROPER(x0))
PROPER(eq(x0, false)) → c19(EQ(proper(x0), ok(false)), PROPER(x0))
PROPER(eq(x0, nil)) → c19(EQ(proper(x0), ok(nil)), PROPER(x0))
PROPER(eq(0, x1)) → c19(EQ(ok(0), proper(x1)), PROPER(x1))
PROPER(eq(true, x1)) → c19(EQ(ok(true), proper(x1)), PROPER(x1))
PROPER(eq(false, x1)) → c19(EQ(ok(false), proper(x1)), PROPER(x1))
PROPER(eq(nil, x1)) → c19(EQ(ok(nil), proper(x1)), PROPER(x1))
PROPER(s(eq(z0, z1))) → c22(S(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(s(s(z0))) → c22(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(inf(z0))) → c22(S(inf(proper(z0))), PROPER(inf(z0)))
PROPER(s(cons(z0, z1))) → c22(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(take(z0, z1))) → c22(S(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(s(length(z0))) → c22(S(length(proper(z0))), PROPER(length(z0)))
PROPER(s(0)) → c22(S(ok(0)))
PROPER(s(true)) → c22(S(ok(true)))
PROPER(s(false)) → c22(S(ok(false)))
PROPER(s(nil)) → c22(S(ok(nil)))
PROPER(inf(eq(z0, z1))) → c24(INF(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(inf(s(z0))) → c24(INF(s(proper(z0))), PROPER(s(z0)))
PROPER(inf(inf(z0))) → c24(INF(inf(proper(z0))), PROPER(inf(z0)))
PROPER(inf(cons(z0, z1))) → c24(INF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(inf(take(z0, z1))) → c24(INF(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(inf(length(z0))) → c24(INF(length(proper(z0))), PROPER(length(z0)))
PROPER(inf(0)) → c24(INF(ok(0)))
PROPER(inf(true)) → c24(INF(ok(true)))
PROPER(inf(false)) → c24(INF(ok(false)))
PROPER(inf(nil)) → c24(INF(ok(nil)))
PROPER(cons(x0, eq(z0, z1))) → c25(CONS(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(cons(x0, s(z0))) → c25(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, inf(z0))) → c25(CONS(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(cons(x0, cons(z0, z1))) → c25(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, take(z0, z1))) → c25(CONS(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(cons(x0, length(z0))) → c25(CONS(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(cons(eq(z0, z1), x1)) → c25(CONS(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c25(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(inf(z0), x1)) → c25(CONS(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c25(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(take(z0, z1), x1)) → c25(CONS(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(cons(length(z0), x1)) → c25(CONS(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c25(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, true)) → c25(CONS(proper(x0), ok(true)), PROPER(x0))
PROPER(cons(x0, false)) → c25(CONS(proper(x0), ok(false)), PROPER(x0))
PROPER(cons(x0, nil)) → c25(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c25(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(true, x1)) → c25(CONS(ok(true), proper(x1)), PROPER(x1))
PROPER(cons(false, x1)) → c25(CONS(ok(false), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c25(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(take(x0, eq(z0, z1))) → c26(TAKE(proper(x0), eq(proper(z0), proper(z1))), PROPER(x0), PROPER(eq(z0, z1)))
PROPER(take(x0, s(z0))) → c26(TAKE(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(take(x0, inf(z0))) → c26(TAKE(proper(x0), inf(proper(z0))), PROPER(x0), PROPER(inf(z0)))
PROPER(take(x0, cons(z0, z1))) → c26(TAKE(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(take(x0, take(z0, z1))) → c26(TAKE(proper(x0), take(proper(z0), proper(z1))), PROPER(x0), PROPER(take(z0, z1)))
PROPER(take(x0, length(z0))) → c26(TAKE(proper(x0), length(proper(z0))), PROPER(x0), PROPER(length(z0)))
PROPER(take(eq(z0, z1), x1)) → c26(TAKE(eq(proper(z0), proper(z1)), proper(x1)), PROPER(eq(z0, z1)), PROPER(x1))
PROPER(take(s(z0), x1)) → c26(TAKE(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(take(inf(z0), x1)) → c26(TAKE(inf(proper(z0)), proper(x1)), PROPER(inf(z0)), PROPER(x1))
PROPER(take(cons(z0, z1), x1)) → c26(TAKE(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(take(take(z0, z1), x1)) → c26(TAKE(take(proper(z0), proper(z1)), proper(x1)), PROPER(take(z0, z1)), PROPER(x1))
PROPER(take(length(z0), x1)) → c26(TAKE(length(proper(z0)), proper(x1)), PROPER(length(z0)), PROPER(x1))
PROPER(take(x0, 0)) → c26(TAKE(proper(x0), ok(0)), PROPER(x0))
PROPER(take(x0, true)) → c26(TAKE(proper(x0), ok(true)), PROPER(x0))
PROPER(take(x0, false)) → c26(TAKE(proper(x0), ok(false)), PROPER(x0))
PROPER(take(x0, nil)) → c26(TAKE(proper(x0), ok(nil)), PROPER(x0))
PROPER(take(0, x1)) → c26(TAKE(ok(0), proper(x1)), PROPER(x1))
PROPER(take(true, x1)) → c26(TAKE(ok(true), proper(x1)), PROPER(x1))
PROPER(take(false, x1)) → c26(TAKE(ok(false), proper(x1)), PROPER(x1))
PROPER(take(nil, x1)) → c26(TAKE(ok(nil), proper(x1)), PROPER(x1))
PROPER(length(eq(z0, z1))) → c28(LENGTH(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
PROPER(length(s(z0))) → c28(LENGTH(s(proper(z0))), PROPER(s(z0)))
PROPER(length(inf(z0))) → c28(LENGTH(inf(proper(z0))), PROPER(inf(z0)))
PROPER(length(cons(z0, z1))) → c28(LENGTH(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(length(take(z0, z1))) → c28(LENGTH(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
PROPER(length(length(z0))) → c28(LENGTH(length(proper(z0))), PROPER(length(z0)))
PROPER(length(0)) → c28(LENGTH(ok(0)))
PROPER(length(true)) → c28(LENGTH(ok(true)))
PROPER(length(false)) → c28(LENGTH(ok(false)))
PROPER(length(nil)) → c28(LENGTH(ok(nil)))
TOP(mark(eq(z0, z1))) → c32(TOP(eq(proper(z0), proper(z1))), PROPER(eq(z0, z1)))
TOP(mark(s(z0))) → c32(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(inf(z0))) → c32(TOP(inf(proper(z0))), PROPER(inf(z0)))
TOP(mark(cons(z0, z1))) → c32(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(take(z0, z1))) → c32(TOP(take(proper(z0), proper(z1))), PROPER(take(z0, z1)))
TOP(mark(length(z0))) → c32(TOP(length(proper(z0))), PROPER(length(z0)))
TOP(ok(eq(s(z0), s(z1)))) → c33(TOP(mark(eq(z0, z1))), ACTIVE(eq(s(z0), s(z1))))
TOP(ok(eq(z0, z1))) → c33(TOP(mark(false)), ACTIVE(eq(z0, z1)))
TOP(ok(inf(z0))) → c33(TOP(mark(cons(z0, inf(s(z0))))), ACTIVE(inf(z0)))
TOP(ok(take(0, z0))) → c33(TOP(mark(nil)), ACTIVE(take(0, z0)))
TOP(ok(take(s(z0), cons(z1, z2)))) → c33(TOP(mark(cons(z1, take(z0, z2)))), ACTIVE(take(s(z0), cons(z1, z2))))
TOP(ok(length(cons(z0, z1)))) → c33(TOP(mark(s(length(z1)))), ACTIVE(length(cons(z0, z1))))
TOP(ok(inf(z0))) → c33(TOP(inf(active(z0))), ACTIVE(inf(z0)))
TOP(ok(take(z0, z1))) → c33(TOP(take(active(z0), z1)), ACTIVE(take(z0, z1)))
TOP(ok(take(z0, z1))) → c33(TOP(take(z0, active(z1))), ACTIVE(take(z0, z1)))
TOP(ok(length(z0))) → c33(TOP(length(active(z0))), ACTIVE(length(z0)))

(62) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
s(ok(z0)) → ok(s(z0))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))

Tuples:

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

S tuples:

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

K tuples:none
Defined Rule Symbols:

active, cons, take, s, length, inf, proper, eq

Defined Pair Symbols:

INF, TAKE, LENGTH, EQ, S, CONS

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c29, c30, c31

(63) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

active(eq(0, 0)) → mark(true)
active(eq(s(z0), s(z1))) → mark(eq(z0, z1))
active(eq(z0, z1)) → mark(false)
active(inf(z0)) → mark(cons(z0, inf(s(z0))))
active(take(0, z0)) → mark(nil)
active(take(s(z0), cons(z1, z2))) → mark(cons(z1, take(z0, z2)))
active(length(nil)) → mark(0)
active(length(cons(z0, z1))) → mark(s(length(z1)))
active(inf(z0)) → inf(active(z0))
active(take(z0, z1)) → take(active(z0), z1)
active(take(z0, z1)) → take(z0, active(z1))
active(length(z0)) → length(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
take(mark(z0), z1) → mark(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
s(ok(z0)) → ok(s(z0))
length(mark(z0)) → mark(length(z0))
length(ok(z0)) → ok(length(z0))
inf(mark(z0)) → mark(inf(z0))
inf(ok(z0)) → ok(inf(z0))
proper(eq(z0, z1)) → eq(proper(z0), proper(z1))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(z0)) → s(proper(z0))
proper(false) → ok(false)
proper(inf(z0)) → inf(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(take(z0, z1)) → take(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(length(z0)) → length(proper(z0))
eq(ok(z0), ok(z1)) → ok(eq(z0, z1))

(64) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

S tuples:

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

INF, TAKE, LENGTH, EQ, S, CONS

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c29, c30, c31

(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.

TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

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

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

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

POL(CONS(x₁, x₂)) = [2]x₁ + [3]x₂
POL(EQ(x₁, x₂)) = [3]x₁ + [3]x₂
POL(INF(x₁)) = 0
POL(LENGTH(x₁)) = 0
POL(S(x₁)) = [3]x₁
POL(TAKE(x₁, x₂)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(ok(x₁)) = [1] + x₁

(66) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

S tuples:

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))

K tuples:

TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

Defined Rule Symbols:none

Defined Pair Symbols:

INF, TAKE, LENGTH, EQ, S, CONS

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c29, c30, c31

(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.

TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))

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

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

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

POL(CONS(x₁, x₂)) = [4]x₁ + [2]x₂
POL(EQ(x₁, x₂)) = [2]x₁ + [4]x₂
POL(INF(x₁)) = 0
POL(LENGTH(x₁)) = 0
POL(S(x₁)) = [4]x₁
POL(TAKE(x₁, x₂)) = [4]x₁ + x₂
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁

(68) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

S tuples:

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))

K tuples:

TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))

Defined Rule Symbols:none

Defined Pair Symbols:

INF, TAKE, LENGTH, EQ, S, CONS

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c29, c30, c31

(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.

INF(mark(z0)) → c12(INF(z0))

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

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

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

POL(CONS(x₁, x₂)) = [2]x₁ + [3]x₂
POL(EQ(x₁, x₂)) = 0
POL(INF(x₁)) = x₁
POL(LENGTH(x₁)) = 0
POL(S(x₁)) = 0
POL(TAKE(x₁, x₂)) = [3]x₁ + [4]x₂
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁

(70) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

S tuples:

INF(ok(z0)) → c13(INF(z0))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))

K tuples:

TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
INF(mark(z0)) → c12(INF(z0))

Defined Rule Symbols:none

Defined Pair Symbols:

INF, TAKE, LENGTH, EQ, S, CONS

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c29, c30, c31

(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.

LENGTH(mark(z0)) → c17(LENGTH(z0))

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

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

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

POL(CONS(x₁, x₂)) = [2]x₁ + [2]x₂
POL(EQ(x₁, x₂)) = x₁ + [2]x₂
POL(INF(x₁)) = 0
POL(LENGTH(x₁)) = x₁
POL(S(x₁)) = [3]x₁
POL(TAKE(x₁, x₂)) = 0
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁

(72) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

S tuples:

INF(ok(z0)) → c13(INF(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))

K tuples:

TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
INF(mark(z0)) → c12(INF(z0))
LENGTH(mark(z0)) → c17(LENGTH(z0))

Defined Rule Symbols:none

Defined Pair Symbols:

INF, TAKE, LENGTH, EQ, S, CONS

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c29, c30, c31

(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.

INF(ok(z0)) → c13(INF(z0))

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

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

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

POL(CONS(x₁, x₂)) = 0
POL(EQ(x₁, x₂)) = [2]x₁ + [4]x₂
POL(INF(x₁)) = [4]x₁
POL(LENGTH(x₁)) = 0
POL(S(x₁)) = 0
POL(TAKE(x₁, x₂)) = [2]x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(ok(x₁)) = [1] + x₁

(74) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

S tuples:

LENGTH(ok(z0)) → c18(LENGTH(z0))

K tuples:

TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
INF(mark(z0)) → c12(INF(z0))
LENGTH(mark(z0)) → c17(LENGTH(z0))
INF(ok(z0)) → c13(INF(z0))

Defined Rule Symbols:none

Defined Pair Symbols:

INF, TAKE, LENGTH, EQ, S, CONS

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c29, c30, c31

(75) 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)) → c18(LENGTH(z0))

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

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

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

POL(CONS(x₁, x₂)) = x₁ + [5]x₂
POL(EQ(x₁, x₂)) = [4]x₂
POL(INF(x₁)) = [2]x₁
POL(LENGTH(x₁)) = [4]x₁
POL(S(x₁)) = [4]x₁
POL(TAKE(x₁, x₂)) = [4]x₂
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(ok(x₁)) = [1] + x₁

(76) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

INF(mark(z0)) → c12(INF(z0))
INF(ok(z0)) → c13(INF(z0))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
LENGTH(mark(z0)) → c17(LENGTH(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))

S tuples:none
K tuples:

TAKE(ok(z0), ok(z1)) → c16(TAKE(z0, z1))
EQ(ok(z0), ok(z1)) → c29(EQ(z0, z1))
S(ok(z0)) → c30(S(z0))
CONS(ok(z0), ok(z1)) → c31(CONS(z0, z1))
TAKE(mark(z0), z1) → c14(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c15(TAKE(z0, z1))
INF(mark(z0)) → c12(INF(z0))
LENGTH(mark(z0)) → c17(LENGTH(z0))
INF(ok(z0)) → c13(INF(z0))
LENGTH(ok(z0)) → c18(LENGTH(z0))

Defined Rule Symbols:none

Defined Pair Symbols:

INF, TAKE, LENGTH, EQ, S, CONS

Compound Symbols:

c12, c13, c14, c15, c16, c17, c18, c29, c30, c31

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

The set S is empty

(0) Obligation:

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

(2) Obligation:

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

(4) Obligation:

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

(6) Obligation:

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

(8) Obligation:

(9) CdtUsableRulesProof (EQUIVALENT transformation)

(10) Obligation:

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

(12) Obligation:

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

(14) Obligation:

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

(16) Obligation:

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

(18) Obligation:

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

(20) Obligation:

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

(22) Obligation:

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

(24) Obligation:

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

(26) Obligation:

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

(28) Obligation:

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

(30) Obligation:

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

(32) Obligation:

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

(34) Obligation:

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

(36) Obligation:

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

(38) Obligation:

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

(40) Obligation:

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

(42) Obligation:

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

(44) Obligation:

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

(46) Obligation:

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

(48) Obligation:

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

(50) Obligation:

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

(52) Obligation:

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

(54) Obligation:

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

(56) Obligation:

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

(58) Obligation:

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

(60) Obligation:

(61) CdtUnreachableProof (EQUIVALENT transformation)

(62) Obligation:

(63) CdtUsableRulesProof (EQUIVALENT transformation)

(64) Obligation:

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

(66) Obligation:

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

(68) Obligation:

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

(70) Obligation:

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

(72) Obligation:

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

(74) Obligation:

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

(76) Obligation:

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

(78) BOUNDS(O(1), O(1))