Runtime Complexity (innermost) proof of /tmp/tmpy8538t/Ex2_Luc03b

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), 86 ms)

↳2 CdtProblem

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

↳4 CdtProblem

↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 2490 ms)

↳6 CdtProblem

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

↳8 CdtProblem

↳9 CdtUnreachableProof (⇔, 0 ms)

↳10 CdtProblem

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

↳12 CdtProblem

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

↳14 CdtProblem

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

↳16 CdtProblem

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

↳18 CdtProblem

↳19 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 26 ms)

↳20 CdtProblem

↳21 CdtUnreachableProof (⇔, 0 ms)

↳22 CdtProblem

↳23 CdtLeafRemovalProof (ComplexityIfPolyImplication, 0 ms)

↳24 CdtProblem

↳25 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 19 ms)

↳26 CdtProblem

↳27 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 0 ms)

↳28 CdtProblem

↳29 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 14 ms)

↳30 CdtProblem

↳31 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 0 ms)

↳32 CdtProblem

↳33 SIsEmptyProof (⇔, 0 ms)

↳34 BOUNDS(O(1), O(1))

(0) Obligation:

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

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

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ACTIVE(fst(s(z0), cons(z1, z2))) → c1(CONS(z1, fst(z0, z2)), FST(z0, z2))
ACTIVE(from(z0)) → c2(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(add(s(z0), z1)) → c4(S(add(z0, z1)), ADD(z0, z1))
ACTIVE(len(cons(z0, z1))) → c6(S(len(z1)), LEN(z1))
ACTIVE(cons(z0, z1)) → c7(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c8(FST(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c9(FST(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c10(FROM(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c12(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(len(z0)) → c13(LEN(active(z0)), ACTIVE(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
FST(mark(z0), z1) → c16(FST(z0, z1))
FST(z0, mark(z1)) → c17(FST(z0, z1))
FST(ok(z0), ok(z1)) → c18(FST(z0, z1))
FROM(mark(z0)) → c19(FROM(z0))
FROM(ok(z0)) → c20(FROM(z0))
ADD(mark(z0), z1) → c21(ADD(z0, z1))
ADD(z0, mark(z1)) → c22(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c23(ADD(z0, z1))
LEN(mark(z0)) → c24(LEN(z0))
LEN(ok(z0)) → c25(LEN(z0))
PROPER(s(z0)) → c27(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c29(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fst(z0, z1)) → c30(FST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c32(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(len(z0)) → c33(LEN(proper(z0)), PROPER(z0))
S(ok(z0)) → c34(S(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(fst(s(z0), cons(z1, z2))) → c1(CONS(z1, fst(z0, z2)), FST(z0, z2))
ACTIVE(from(z0)) → c2(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0))
ACTIVE(add(s(z0), z1)) → c4(S(add(z0, z1)), ADD(z0, z1))
ACTIVE(len(cons(z0, z1))) → c6(S(len(z1)), LEN(z1))
ACTIVE(cons(z0, z1)) → c7(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c8(FST(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c9(FST(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c10(FROM(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c12(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(len(z0)) → c13(LEN(active(z0)), ACTIVE(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
FST(mark(z0), z1) → c16(FST(z0, z1))
FST(z0, mark(z1)) → c17(FST(z0, z1))
FST(ok(z0), ok(z1)) → c18(FST(z0, z1))
FROM(mark(z0)) → c19(FROM(z0))
FROM(ok(z0)) → c20(FROM(z0))
ADD(mark(z0), z1) → c21(ADD(z0, z1))
ADD(z0, mark(z1)) → c22(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c23(ADD(z0, z1))
LEN(mark(z0)) → c24(LEN(z0))
LEN(ok(z0)) → c25(LEN(z0))
PROPER(s(z0)) → c27(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c29(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fst(z0, z1)) → c30(FST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c32(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(len(z0)) → c33(LEN(proper(z0)), PROPER(z0))
S(ok(z0)) → c34(S(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, cons, fst, from, add, len, proper, s, top

Defined Pair Symbols:

ACTIVE, CONS, FST, FROM, ADD, LEN, PROPER, S, TOP

Compound Symbols:

c1, c2, c4, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c27, c29, c30, c31, c32, c33, c34, c35, c36

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

Removed 1 trailing tuple parts

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ACTIVE(fst(s(z0), cons(z1, z2))) → c1(CONS(z1, fst(z0, z2)), FST(z0, z2))
ACTIVE(add(s(z0), z1)) → c4(S(add(z0, z1)), ADD(z0, z1))
ACTIVE(len(cons(z0, z1))) → c6(S(len(z1)), LEN(z1))
ACTIVE(cons(z0, z1)) → c7(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c8(FST(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c9(FST(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c10(FROM(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c12(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(len(z0)) → c13(LEN(active(z0)), ACTIVE(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
FST(mark(z0), z1) → c16(FST(z0, z1))
FST(z0, mark(z1)) → c17(FST(z0, z1))
FST(ok(z0), ok(z1)) → c18(FST(z0, z1))
FROM(mark(z0)) → c19(FROM(z0))
FROM(ok(z0)) → c20(FROM(z0))
ADD(mark(z0), z1) → c21(ADD(z0, z1))
ADD(z0, mark(z1)) → c22(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c23(ADD(z0, z1))
LEN(mark(z0)) → c24(LEN(z0))
LEN(ok(z0)) → c25(LEN(z0))
PROPER(s(z0)) → c27(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c29(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fst(z0, z1)) → c30(FST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c32(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(len(z0)) → c33(LEN(proper(z0)), PROPER(z0))
S(ok(z0)) → c34(S(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c2(CONS(z0, from(s(z0))), S(z0))

S tuples:

ACTIVE(fst(s(z0), cons(z1, z2))) → c1(CONS(z1, fst(z0, z2)), FST(z0, z2))
ACTIVE(add(s(z0), z1)) → c4(S(add(z0, z1)), ADD(z0, z1))
ACTIVE(len(cons(z0, z1))) → c6(S(len(z1)), LEN(z1))
ACTIVE(cons(z0, z1)) → c7(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c8(FST(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c9(FST(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c10(FROM(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c12(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(len(z0)) → c13(LEN(active(z0)), ACTIVE(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
FST(mark(z0), z1) → c16(FST(z0, z1))
FST(z0, mark(z1)) → c17(FST(z0, z1))
FST(ok(z0), ok(z1)) → c18(FST(z0, z1))
FROM(mark(z0)) → c19(FROM(z0))
FROM(ok(z0)) → c20(FROM(z0))
ADD(mark(z0), z1) → c21(ADD(z0, z1))
ADD(z0, mark(z1)) → c22(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c23(ADD(z0, z1))
LEN(mark(z0)) → c24(LEN(z0))
LEN(ok(z0)) → c25(LEN(z0))
PROPER(s(z0)) → c27(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c29(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fst(z0, z1)) → c30(FST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c32(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(len(z0)) → c33(LEN(proper(z0)), PROPER(z0))
S(ok(z0)) → c34(S(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c2(CONS(z0, from(s(z0))), S(z0))

K tuples:none
Defined Rule Symbols:

active, cons, fst, from, add, len, proper, s, top

Defined Pair Symbols:

ACTIVE, CONS, FST, FROM, ADD, LEN, PROPER, S, TOP

Compound Symbols:

c1, c4, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c27, c29, c30, c31, c32, c33, c34, c35, c36, c2

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

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

TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))

We considered the (Usable) Rules:

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

And the Tuples:

ACTIVE(fst(s(z0), cons(z1, z2))) → c1(CONS(z1, fst(z0, z2)), FST(z0, z2))
ACTIVE(add(s(z0), z1)) → c4(S(add(z0, z1)), ADD(z0, z1))
ACTIVE(len(cons(z0, z1))) → c6(S(len(z1)), LEN(z1))
ACTIVE(cons(z0, z1)) → c7(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c8(FST(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c9(FST(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c10(FROM(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c12(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(len(z0)) → c13(LEN(active(z0)), ACTIVE(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
FST(mark(z0), z1) → c16(FST(z0, z1))
FST(z0, mark(z1)) → c17(FST(z0, z1))
FST(ok(z0), ok(z1)) → c18(FST(z0, z1))
FROM(mark(z0)) → c19(FROM(z0))
FROM(ok(z0)) → c20(FROM(z0))
ADD(mark(z0), z1) → c21(ADD(z0, z1))
ADD(z0, mark(z1)) → c22(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c23(ADD(z0, z1))
LEN(mark(z0)) → c24(LEN(z0))
LEN(ok(z0)) → c25(LEN(z0))
PROPER(s(z0)) → c27(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c29(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fst(z0, z1)) → c30(FST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c32(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(len(z0)) → c33(LEN(proper(z0)), PROPER(z0))
S(ok(z0)) → c34(S(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c2(CONS(z0, from(s(z0))), S(z0))

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

POL(0) = 0
POL(ACTIVE(x₁)) = 0
POL(ADD(x₁, x₂)) = 0
POL(CONS(x₁, x₂)) = 0
POL(FROM(x₁)) = 0
POL(FST(x₁, x₂)) = 0
POL(LEN(x₁)) = 0
POL(PROPER(x₁)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = [4]x₁
POL(active(x₁)) = x₁
POL(add(x₁, x₂)) = [2] + [2]x₁ + [2]x₂
POL(c1(x₁, x₂)) = x₁ + x₂
POL(c10(x₁, x₂)) = x₁ + x₂
POL(c11(x₁, x₂)) = x₁ + x₂
POL(c12(x₁, x₂)) = x₁ + x₂
POL(c13(x₁, x₂)) = x₁ + x₂
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c2(x₁, x₂)) = x₁ + x₂
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c27(x₁, x₂)) = x₁ + x₂
POL(c29(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c30(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c31(x₁, x₂)) = x₁ + x₂
POL(c32(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c33(x₁, x₂)) = x₁ + x₂
POL(c34(x₁)) = x₁
POL(c35(x₁, x₂)) = x₁ + x₂
POL(c36(x₁, x₂)) = x₁ + x₂
POL(c4(x₁, x₂)) = x₁ + x₂
POL(c6(x₁, x₂)) = x₁ + x₂
POL(c7(x₁, x₂)) = x₁ + x₂
POL(c8(x₁, x₂)) = x₁ + x₂
POL(c9(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = [2] + x₁
POL(from(x₁)) = [4] + x₁
POL(fst(x₁, x₂)) = [2] + [4]x₁ + [2]x₂
POL(len(x₁)) = [4]x₁
POL(mark(x₁)) = [1] + x₁
POL(nil) = [1]
POL(ok(x₁)) = x₁
POL(proper(x₁)) = x₁
POL(s(x₁)) = 0

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ACTIVE(fst(s(z0), cons(z1, z2))) → c1(CONS(z1, fst(z0, z2)), FST(z0, z2))
ACTIVE(add(s(z0), z1)) → c4(S(add(z0, z1)), ADD(z0, z1))
ACTIVE(len(cons(z0, z1))) → c6(S(len(z1)), LEN(z1))
ACTIVE(cons(z0, z1)) → c7(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c8(FST(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c9(FST(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c10(FROM(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c12(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(len(z0)) → c13(LEN(active(z0)), ACTIVE(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
FST(mark(z0), z1) → c16(FST(z0, z1))
FST(z0, mark(z1)) → c17(FST(z0, z1))
FST(ok(z0), ok(z1)) → c18(FST(z0, z1))
FROM(mark(z0)) → c19(FROM(z0))
FROM(ok(z0)) → c20(FROM(z0))
ADD(mark(z0), z1) → c21(ADD(z0, z1))
ADD(z0, mark(z1)) → c22(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c23(ADD(z0, z1))
LEN(mark(z0)) → c24(LEN(z0))
LEN(ok(z0)) → c25(LEN(z0))
PROPER(s(z0)) → c27(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c29(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fst(z0, z1)) → c30(FST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c32(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(len(z0)) → c33(LEN(proper(z0)), PROPER(z0))
S(ok(z0)) → c34(S(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c2(CONS(z0, from(s(z0))), S(z0))

S tuples:

ACTIVE(fst(s(z0), cons(z1, z2))) → c1(CONS(z1, fst(z0, z2)), FST(z0, z2))
ACTIVE(add(s(z0), z1)) → c4(S(add(z0, z1)), ADD(z0, z1))
ACTIVE(len(cons(z0, z1))) → c6(S(len(z1)), LEN(z1))
ACTIVE(cons(z0, z1)) → c7(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c8(FST(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c9(FST(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c10(FROM(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c12(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(len(z0)) → c13(LEN(active(z0)), ACTIVE(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
FST(mark(z0), z1) → c16(FST(z0, z1))
FST(z0, mark(z1)) → c17(FST(z0, z1))
FST(ok(z0), ok(z1)) → c18(FST(z0, z1))
FROM(mark(z0)) → c19(FROM(z0))
FROM(ok(z0)) → c20(FROM(z0))
ADD(mark(z0), z1) → c21(ADD(z0, z1))
ADD(z0, mark(z1)) → c22(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c23(ADD(z0, z1))
LEN(mark(z0)) → c24(LEN(z0))
LEN(ok(z0)) → c25(LEN(z0))
PROPER(s(z0)) → c27(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c29(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fst(z0, z1)) → c30(FST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c32(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(len(z0)) → c33(LEN(proper(z0)), PROPER(z0))
S(ok(z0)) → c34(S(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c2(CONS(z0, from(s(z0))), S(z0))

K tuples:

TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, cons, fst, from, add, len, proper, s, top

Defined Pair Symbols:

ACTIVE, CONS, FST, FROM, ADD, LEN, PROPER, S, TOP

Compound Symbols:

c1, c4, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c27, c29, c30, c31, c32, c33, c34, c35, c36, c2

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

Use narrowing to replace ACTIVE(add(s(z0), z1)) → c4(S(add(z0, z1)), ADD(z0, z1)) by

ACTIVE(add(s(mark(z0)), z1)) → c4(S(mark(add(z0, z1))), ADD(mark(z0), z1))
ACTIVE(add(s(ok(z0)), ok(z1))) → c4(S(ok(add(z0, z1))), ADD(ok(z0), ok(z1)))
ACTIVE(add(s(x0), x1)) → c4(ADD(x0, x1))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ACTIVE(fst(s(z0), cons(z1, z2))) → c1(CONS(z1, fst(z0, z2)), FST(z0, z2))
ACTIVE(len(cons(z0, z1))) → c6(S(len(z1)), LEN(z1))
ACTIVE(cons(z0, z1)) → c7(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c8(FST(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c9(FST(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c10(FROM(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c12(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(len(z0)) → c13(LEN(active(z0)), ACTIVE(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
FST(mark(z0), z1) → c16(FST(z0, z1))
FST(z0, mark(z1)) → c17(FST(z0, z1))
FST(ok(z0), ok(z1)) → c18(FST(z0, z1))
FROM(mark(z0)) → c19(FROM(z0))
FROM(ok(z0)) → c20(FROM(z0))
ADD(mark(z0), z1) → c21(ADD(z0, z1))
ADD(z0, mark(z1)) → c22(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c23(ADD(z0, z1))
LEN(mark(z0)) → c24(LEN(z0))
LEN(ok(z0)) → c25(LEN(z0))
PROPER(s(z0)) → c27(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c29(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fst(z0, z1)) → c30(FST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c32(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(len(z0)) → c33(LEN(proper(z0)), PROPER(z0))
S(ok(z0)) → c34(S(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c2(CONS(z0, from(s(z0))), S(z0))
ACTIVE(add(s(mark(z0)), z1)) → c4(S(mark(add(z0, z1))), ADD(mark(z0), z1))
ACTIVE(add(s(ok(z0)), ok(z1))) → c4(S(ok(add(z0, z1))), ADD(ok(z0), ok(z1)))
ACTIVE(add(s(x0), x1)) → c4(ADD(x0, x1))

S tuples:

ACTIVE(fst(s(z0), cons(z1, z2))) → c1(CONS(z1, fst(z0, z2)), FST(z0, z2))
ACTIVE(len(cons(z0, z1))) → c6(S(len(z1)), LEN(z1))
ACTIVE(cons(z0, z1)) → c7(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c8(FST(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c9(FST(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c10(FROM(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c12(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(len(z0)) → c13(LEN(active(z0)), ACTIVE(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
FST(mark(z0), z1) → c16(FST(z0, z1))
FST(z0, mark(z1)) → c17(FST(z0, z1))
FST(ok(z0), ok(z1)) → c18(FST(z0, z1))
FROM(mark(z0)) → c19(FROM(z0))
FROM(ok(z0)) → c20(FROM(z0))
ADD(mark(z0), z1) → c21(ADD(z0, z1))
ADD(z0, mark(z1)) → c22(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c23(ADD(z0, z1))
LEN(mark(z0)) → c24(LEN(z0))
LEN(ok(z0)) → c25(LEN(z0))
PROPER(s(z0)) → c27(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c29(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fst(z0, z1)) → c30(FST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c32(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(len(z0)) → c33(LEN(proper(z0)), PROPER(z0))
S(ok(z0)) → c34(S(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c2(CONS(z0, from(s(z0))), S(z0))
ACTIVE(add(s(mark(z0)), z1)) → c4(S(mark(add(z0, z1))), ADD(mark(z0), z1))
ACTIVE(add(s(ok(z0)), ok(z1))) → c4(S(ok(add(z0, z1))), ADD(ok(z0), ok(z1)))
ACTIVE(add(s(x0), x1)) → c4(ADD(x0, x1))

K tuples:

TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, cons, fst, from, add, len, proper, s, top

Defined Pair Symbols:

ACTIVE, CONS, FST, FROM, ADD, LEN, PROPER, S, TOP

Compound Symbols:

c1, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c27, c29, c30, c31, c32, c33, c34, c35, c36, c2, c4, c4

(9) CdtUnreachableProof (EQUIVALENT transformation)

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

ACTIVE(add(s(ok(z0)), ok(z1))) → c4(S(ok(add(z0, z1))), ADD(ok(z0), ok(z1)))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
FST(mark(z0), z1) → c16(FST(z0, z1))
FST(z0, mark(z1)) → c17(FST(z0, z1))
FST(ok(z0), ok(z1)) → c18(FST(z0, z1))
FROM(mark(z0)) → c19(FROM(z0))
FROM(ok(z0)) → c20(FROM(z0))
ADD(mark(z0), z1) → c21(ADD(z0, z1))
ADD(z0, mark(z1)) → c22(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c23(ADD(z0, z1))
LEN(mark(z0)) → c24(LEN(z0))
LEN(ok(z0)) → c25(LEN(z0))
S(ok(z0)) → c34(S(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fst(s(z0), cons(z1, z2))) → c1(CONS(z1, fst(z0, z2)), FST(z0, z2))
ACTIVE(len(cons(z0, z1))) → c6(S(len(z1)), LEN(z1))
ACTIVE(cons(z0, z1)) → c7(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c8(FST(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c9(FST(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c10(FROM(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c12(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(len(z0)) → c13(LEN(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c2(CONS(z0, from(s(z0))), S(z0))
ACTIVE(add(s(mark(z0)), z1)) → c4(S(mark(add(z0, z1))), ADD(mark(z0), z1))
ACTIVE(add(s(x0), x1)) → c4(ADD(x0, x1))
PROPER(s(z0)) → c27(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c29(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fst(z0, z1)) → c30(FST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c32(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(len(z0)) → c33(LEN(proper(z0)), PROPER(z0))

S tuples:

ACTIVE(fst(s(z0), cons(z1, z2))) → c1(CONS(z1, fst(z0, z2)), FST(z0, z2))
ACTIVE(len(cons(z0, z1))) → c6(S(len(z1)), LEN(z1))
ACTIVE(cons(z0, z1)) → c7(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c8(FST(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c9(FST(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c10(FROM(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c12(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(len(z0)) → c13(LEN(active(z0)), ACTIVE(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
FST(mark(z0), z1) → c16(FST(z0, z1))
FST(z0, mark(z1)) → c17(FST(z0, z1))
FST(ok(z0), ok(z1)) → c18(FST(z0, z1))
FROM(mark(z0)) → c19(FROM(z0))
FROM(ok(z0)) → c20(FROM(z0))
ADD(mark(z0), z1) → c21(ADD(z0, z1))
ADD(z0, mark(z1)) → c22(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c23(ADD(z0, z1))
LEN(mark(z0)) → c24(LEN(z0))
LEN(ok(z0)) → c25(LEN(z0))
PROPER(s(z0)) → c27(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c29(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fst(z0, z1)) → c30(FST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c32(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(len(z0)) → c33(LEN(proper(z0)), PROPER(z0))
S(ok(z0)) → c34(S(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c2(CONS(z0, from(s(z0))), S(z0))
ACTIVE(add(s(mark(z0)), z1)) → c4(S(mark(add(z0, z1))), ADD(mark(z0), z1))
ACTIVE(add(s(x0), x1)) → c4(ADD(x0, x1))

K tuples:

TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, cons, fst, from, add, len, proper, s, top

Defined Pair Symbols:

CONS, FST, FROM, ADD, LEN, S, TOP, ACTIVE, PROPER

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c34, c35, c36, c1, c6, c7, c8, c9, c10, c11, c12, c13, c2, c4, c4, c27, c29, c30, c31, c32, c33

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

Removed 1 trailing tuple parts

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
FST(mark(z0), z1) → c16(FST(z0, z1))
FST(z0, mark(z1)) → c17(FST(z0, z1))
FST(ok(z0), ok(z1)) → c18(FST(z0, z1))
FROM(mark(z0)) → c19(FROM(z0))
FROM(ok(z0)) → c20(FROM(z0))
ADD(mark(z0), z1) → c21(ADD(z0, z1))
ADD(z0, mark(z1)) → c22(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c23(ADD(z0, z1))
LEN(mark(z0)) → c24(LEN(z0))
LEN(ok(z0)) → c25(LEN(z0))
S(ok(z0)) → c34(S(z0))
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fst(s(z0), cons(z1, z2))) → c1(CONS(z1, fst(z0, z2)), FST(z0, z2))
ACTIVE(len(cons(z0, z1))) → c6(S(len(z1)), LEN(z1))
ACTIVE(cons(z0, z1)) → c7(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c8(FST(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c9(FST(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c10(FROM(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c12(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(len(z0)) → c13(LEN(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c2(CONS(z0, from(s(z0))), S(z0))
ACTIVE(add(s(x0), x1)) → c4(ADD(x0, x1))
PROPER(s(z0)) → c27(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c29(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fst(z0, z1)) → c30(FST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c32(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(len(z0)) → c33(LEN(proper(z0)), PROPER(z0))
ACTIVE(add(s(mark(z0)), z1)) → c4(ADD(mark(z0), z1))

S tuples:

ACTIVE(fst(s(z0), cons(z1, z2))) → c1(CONS(z1, fst(z0, z2)), FST(z0, z2))
ACTIVE(len(cons(z0, z1))) → c6(S(len(z1)), LEN(z1))
ACTIVE(cons(z0, z1)) → c7(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c8(FST(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c9(FST(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c10(FROM(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c12(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(len(z0)) → c13(LEN(active(z0)), ACTIVE(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
FST(mark(z0), z1) → c16(FST(z0, z1))
FST(z0, mark(z1)) → c17(FST(z0, z1))
FST(ok(z0), ok(z1)) → c18(FST(z0, z1))
FROM(mark(z0)) → c19(FROM(z0))
FROM(ok(z0)) → c20(FROM(z0))
ADD(mark(z0), z1) → c21(ADD(z0, z1))
ADD(z0, mark(z1)) → c22(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c23(ADD(z0, z1))
LEN(mark(z0)) → c24(LEN(z0))
LEN(ok(z0)) → c25(LEN(z0))
PROPER(s(z0)) → c27(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c29(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fst(z0, z1)) → c30(FST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c32(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(len(z0)) → c33(LEN(proper(z0)), PROPER(z0))
S(ok(z0)) → c34(S(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c2(CONS(z0, from(s(z0))), S(z0))
ACTIVE(add(s(x0), x1)) → c4(ADD(x0, x1))
ACTIVE(add(s(mark(z0)), z1)) → c4(ADD(mark(z0), z1))

K tuples:

TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, cons, fst, from, add, len, proper, s, top

Defined Pair Symbols:

CONS, FST, FROM, ADD, LEN, S, TOP, ACTIVE, PROPER

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c34, c35, c36, c1, c6, c7, c8, c9, c10, c11, c12, c13, c2, c4, c27, c29, c30, c31, c32, c33

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

Use narrowing to replace TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(0)) → c35(TOP(ok(0)), PROPER(0))
TOP(mark(s(z0))) → c35(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(nil)) → c35(TOP(ok(nil)), PROPER(nil))
TOP(mark(cons(z0, z1))) → c35(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(fst(z0, z1))) → c35(TOP(fst(proper(z0), proper(z1))), PROPER(fst(z0, z1)))
TOP(mark(from(z0))) → c35(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(add(z0, z1))) → c35(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
TOP(mark(len(z0))) → c35(TOP(len(proper(z0))), PROPER(len(z0)))
TOP(mark(x0)) → c35

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
FST(mark(z0), z1) → c16(FST(z0, z1))
FST(z0, mark(z1)) → c17(FST(z0, z1))
FST(ok(z0), ok(z1)) → c18(FST(z0, z1))
FROM(mark(z0)) → c19(FROM(z0))
FROM(ok(z0)) → c20(FROM(z0))
ADD(mark(z0), z1) → c21(ADD(z0, z1))
ADD(z0, mark(z1)) → c22(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c23(ADD(z0, z1))
LEN(mark(z0)) → c24(LEN(z0))
LEN(ok(z0)) → c25(LEN(z0))
S(ok(z0)) → c34(S(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fst(s(z0), cons(z1, z2))) → c1(CONS(z1, fst(z0, z2)), FST(z0, z2))
ACTIVE(len(cons(z0, z1))) → c6(S(len(z1)), LEN(z1))
ACTIVE(cons(z0, z1)) → c7(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c8(FST(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c9(FST(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c10(FROM(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c12(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(len(z0)) → c13(LEN(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c2(CONS(z0, from(s(z0))), S(z0))
ACTIVE(add(s(x0), x1)) → c4(ADD(x0, x1))
PROPER(s(z0)) → c27(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c29(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fst(z0, z1)) → c30(FST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c32(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(len(z0)) → c33(LEN(proper(z0)), PROPER(z0))
ACTIVE(add(s(mark(z0)), z1)) → c4(ADD(mark(z0), z1))
TOP(mark(0)) → c35(TOP(ok(0)), PROPER(0))
TOP(mark(s(z0))) → c35(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(nil)) → c35(TOP(ok(nil)), PROPER(nil))
TOP(mark(cons(z0, z1))) → c35(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(fst(z0, z1))) → c35(TOP(fst(proper(z0), proper(z1))), PROPER(fst(z0, z1)))
TOP(mark(from(z0))) → c35(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(add(z0, z1))) → c35(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
TOP(mark(len(z0))) → c35(TOP(len(proper(z0))), PROPER(len(z0)))
TOP(mark(x0)) → c35

S tuples:

ACTIVE(fst(s(z0), cons(z1, z2))) → c1(CONS(z1, fst(z0, z2)), FST(z0, z2))
ACTIVE(len(cons(z0, z1))) → c6(S(len(z1)), LEN(z1))
ACTIVE(cons(z0, z1)) → c7(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c8(FST(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c9(FST(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c10(FROM(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c12(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(len(z0)) → c13(LEN(active(z0)), ACTIVE(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
FST(mark(z0), z1) → c16(FST(z0, z1))
FST(z0, mark(z1)) → c17(FST(z0, z1))
FST(ok(z0), ok(z1)) → c18(FST(z0, z1))
FROM(mark(z0)) → c19(FROM(z0))
FROM(ok(z0)) → c20(FROM(z0))
ADD(mark(z0), z1) → c21(ADD(z0, z1))
ADD(z0, mark(z1)) → c22(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c23(ADD(z0, z1))
LEN(mark(z0)) → c24(LEN(z0))
LEN(ok(z0)) → c25(LEN(z0))
PROPER(s(z0)) → c27(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c29(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fst(z0, z1)) → c30(FST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c32(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(len(z0)) → c33(LEN(proper(z0)), PROPER(z0))
S(ok(z0)) → c34(S(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c2(CONS(z0, from(s(z0))), S(z0))
ACTIVE(add(s(x0), x1)) → c4(ADD(x0, x1))
ACTIVE(add(s(mark(z0)), z1)) → c4(ADD(mark(z0), z1))

K tuples:

TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, cons, fst, from, add, len, proper, s, top

Defined Pair Symbols:

CONS, FST, FROM, ADD, LEN, S, TOP, ACTIVE, PROPER

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c34, c36, c1, c6, c7, c8, c9, c10, c11, c12, c13, c2, c4, c27, c29, c30, c31, c32, c33, c35, c35

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

Removed 1 trailing nodes:

TOP(mark(x0)) → c35

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
FST(mark(z0), z1) → c16(FST(z0, z1))
FST(z0, mark(z1)) → c17(FST(z0, z1))
FST(ok(z0), ok(z1)) → c18(FST(z0, z1))
FROM(mark(z0)) → c19(FROM(z0))
FROM(ok(z0)) → c20(FROM(z0))
ADD(mark(z0), z1) → c21(ADD(z0, z1))
ADD(z0, mark(z1)) → c22(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c23(ADD(z0, z1))
LEN(mark(z0)) → c24(LEN(z0))
LEN(ok(z0)) → c25(LEN(z0))
S(ok(z0)) → c34(S(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fst(s(z0), cons(z1, z2))) → c1(CONS(z1, fst(z0, z2)), FST(z0, z2))
ACTIVE(len(cons(z0, z1))) → c6(S(len(z1)), LEN(z1))
ACTIVE(cons(z0, z1)) → c7(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c8(FST(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c9(FST(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c10(FROM(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c12(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(len(z0)) → c13(LEN(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c2(CONS(z0, from(s(z0))), S(z0))
ACTIVE(add(s(x0), x1)) → c4(ADD(x0, x1))
PROPER(s(z0)) → c27(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c29(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fst(z0, z1)) → c30(FST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c32(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(len(z0)) → c33(LEN(proper(z0)), PROPER(z0))
ACTIVE(add(s(mark(z0)), z1)) → c4(ADD(mark(z0), z1))
TOP(mark(0)) → c35(TOP(ok(0)), PROPER(0))
TOP(mark(s(z0))) → c35(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(nil)) → c35(TOP(ok(nil)), PROPER(nil))
TOP(mark(cons(z0, z1))) → c35(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(fst(z0, z1))) → c35(TOP(fst(proper(z0), proper(z1))), PROPER(fst(z0, z1)))
TOP(mark(from(z0))) → c35(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(add(z0, z1))) → c35(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
TOP(mark(len(z0))) → c35(TOP(len(proper(z0))), PROPER(len(z0)))

S tuples:

ACTIVE(fst(s(z0), cons(z1, z2))) → c1(CONS(z1, fst(z0, z2)), FST(z0, z2))
ACTIVE(len(cons(z0, z1))) → c6(S(len(z1)), LEN(z1))
ACTIVE(cons(z0, z1)) → c7(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c8(FST(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c9(FST(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c10(FROM(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c12(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(len(z0)) → c13(LEN(active(z0)), ACTIVE(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
FST(mark(z0), z1) → c16(FST(z0, z1))
FST(z0, mark(z1)) → c17(FST(z0, z1))
FST(ok(z0), ok(z1)) → c18(FST(z0, z1))
FROM(mark(z0)) → c19(FROM(z0))
FROM(ok(z0)) → c20(FROM(z0))
ADD(mark(z0), z1) → c21(ADD(z0, z1))
ADD(z0, mark(z1)) → c22(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c23(ADD(z0, z1))
LEN(mark(z0)) → c24(LEN(z0))
LEN(ok(z0)) → c25(LEN(z0))
PROPER(s(z0)) → c27(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c29(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fst(z0, z1)) → c30(FST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c32(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(len(z0)) → c33(LEN(proper(z0)), PROPER(z0))
S(ok(z0)) → c34(S(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c2(CONS(z0, from(s(z0))), S(z0))
ACTIVE(add(s(x0), x1)) → c4(ADD(x0, x1))
ACTIVE(add(s(mark(z0)), z1)) → c4(ADD(mark(z0), z1))

K tuples:none
Defined Rule Symbols:

active, cons, fst, from, add, len, proper, s, top

Defined Pair Symbols:

CONS, FST, FROM, ADD, LEN, S, TOP, ACTIVE, PROPER

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c34, c36, c1, c6, c7, c8, c9, c10, c11, c12, c13, c2, c4, c27, c29, c30, c31, c32, c33, c35

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

Removed 2 trailing tuple parts

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
FST(mark(z0), z1) → c16(FST(z0, z1))
FST(z0, mark(z1)) → c17(FST(z0, z1))
FST(ok(z0), ok(z1)) → c18(FST(z0, z1))
FROM(mark(z0)) → c19(FROM(z0))
FROM(ok(z0)) → c20(FROM(z0))
ADD(mark(z0), z1) → c21(ADD(z0, z1))
ADD(z0, mark(z1)) → c22(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c23(ADD(z0, z1))
LEN(mark(z0)) → c24(LEN(z0))
LEN(ok(z0)) → c25(LEN(z0))
S(ok(z0)) → c34(S(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(fst(s(z0), cons(z1, z2))) → c1(CONS(z1, fst(z0, z2)), FST(z0, z2))
ACTIVE(len(cons(z0, z1))) → c6(S(len(z1)), LEN(z1))
ACTIVE(cons(z0, z1)) → c7(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c8(FST(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c9(FST(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c10(FROM(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c12(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(len(z0)) → c13(LEN(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c2(CONS(z0, from(s(z0))), S(z0))
ACTIVE(add(s(x0), x1)) → c4(ADD(x0, x1))
PROPER(s(z0)) → c27(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c29(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fst(z0, z1)) → c30(FST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c32(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(len(z0)) → c33(LEN(proper(z0)), PROPER(z0))
ACTIVE(add(s(mark(z0)), z1)) → c4(ADD(mark(z0), z1))
TOP(mark(s(z0))) → c35(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(cons(z0, z1))) → c35(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(fst(z0, z1))) → c35(TOP(fst(proper(z0), proper(z1))), PROPER(fst(z0, z1)))
TOP(mark(from(z0))) → c35(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(add(z0, z1))) → c35(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
TOP(mark(len(z0))) → c35(TOP(len(proper(z0))), PROPER(len(z0)))
TOP(mark(0)) → c35(TOP(ok(0)))
TOP(mark(nil)) → c35(TOP(ok(nil)))

S tuples:

ACTIVE(fst(s(z0), cons(z1, z2))) → c1(CONS(z1, fst(z0, z2)), FST(z0, z2))
ACTIVE(len(cons(z0, z1))) → c6(S(len(z1)), LEN(z1))
ACTIVE(cons(z0, z1)) → c7(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c8(FST(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c9(FST(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c10(FROM(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c12(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(len(z0)) → c13(LEN(active(z0)), ACTIVE(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
FST(mark(z0), z1) → c16(FST(z0, z1))
FST(z0, mark(z1)) → c17(FST(z0, z1))
FST(ok(z0), ok(z1)) → c18(FST(z0, z1))
FROM(mark(z0)) → c19(FROM(z0))
FROM(ok(z0)) → c20(FROM(z0))
ADD(mark(z0), z1) → c21(ADD(z0, z1))
ADD(z0, mark(z1)) → c22(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c23(ADD(z0, z1))
LEN(mark(z0)) → c24(LEN(z0))
LEN(ok(z0)) → c25(LEN(z0))
PROPER(s(z0)) → c27(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c29(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fst(z0, z1)) → c30(FST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c32(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(len(z0)) → c33(LEN(proper(z0)), PROPER(z0))
S(ok(z0)) → c34(S(z0))
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c2(CONS(z0, from(s(z0))), S(z0))
ACTIVE(add(s(x0), x1)) → c4(ADD(x0, x1))
ACTIVE(add(s(mark(z0)), z1)) → c4(ADD(mark(z0), z1))

K tuples:none
Defined Rule Symbols:

active, cons, fst, from, add, len, proper, s, top

Defined Pair Symbols:

CONS, FST, FROM, ADD, LEN, S, TOP, ACTIVE, PROPER

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c34, c36, c1, c6, c7, c8, c9, c10, c11, c12, c13, c2, c4, c27, c29, c30, c31, c32, c33, c35, c35

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

Use narrowing to replace TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(fst(0, z0))) → c36(TOP(mark(nil)), ACTIVE(fst(0, z0)))
TOP(ok(fst(s(z0), cons(z1, z2)))) → c36(TOP(mark(cons(z1, fst(z0, z2)))), ACTIVE(fst(s(z0), cons(z1, z2))))
TOP(ok(from(z0))) → c36(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(add(0, z0))) → c36(TOP(mark(z0)), ACTIVE(add(0, z0)))
TOP(ok(add(s(z0), z1))) → c36(TOP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
TOP(ok(len(nil))) → c36(TOP(mark(0)), ACTIVE(len(nil)))
TOP(ok(len(cons(z0, z1)))) → c36(TOP(mark(s(len(z1)))), ACTIVE(len(cons(z0, z1))))
TOP(ok(cons(z0, z1))) → c36(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(fst(z0, z1))) → c36(TOP(fst(active(z0), z1)), ACTIVE(fst(z0, z1)))
TOP(ok(fst(z0, z1))) → c36(TOP(fst(z0, active(z1))), ACTIVE(fst(z0, z1)))
TOP(ok(from(z0))) → c36(TOP(from(active(z0))), ACTIVE(from(z0)))
TOP(ok(add(z0, z1))) → c36(TOP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
TOP(ok(add(z0, z1))) → c36(TOP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
TOP(ok(len(z0))) → c36(TOP(len(active(z0))), ACTIVE(len(z0)))
TOP(ok(x0)) → c36

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
FST(mark(z0), z1) → c16(FST(z0, z1))
FST(z0, mark(z1)) → c17(FST(z0, z1))
FST(ok(z0), ok(z1)) → c18(FST(z0, z1))
FROM(mark(z0)) → c19(FROM(z0))
FROM(ok(z0)) → c20(FROM(z0))
ADD(mark(z0), z1) → c21(ADD(z0, z1))
ADD(z0, mark(z1)) → c22(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c23(ADD(z0, z1))
LEN(mark(z0)) → c24(LEN(z0))
LEN(ok(z0)) → c25(LEN(z0))
S(ok(z0)) → c34(S(z0))
ACTIVE(fst(s(z0), cons(z1, z2))) → c1(CONS(z1, fst(z0, z2)), FST(z0, z2))
ACTIVE(len(cons(z0, z1))) → c6(S(len(z1)), LEN(z1))
ACTIVE(cons(z0, z1)) → c7(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c8(FST(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c9(FST(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c10(FROM(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c12(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(len(z0)) → c13(LEN(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c2(CONS(z0, from(s(z0))), S(z0))
ACTIVE(add(s(x0), x1)) → c4(ADD(x0, x1))
PROPER(s(z0)) → c27(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c29(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fst(z0, z1)) → c30(FST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c32(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(len(z0)) → c33(LEN(proper(z0)), PROPER(z0))
ACTIVE(add(s(mark(z0)), z1)) → c4(ADD(mark(z0), z1))
TOP(mark(s(z0))) → c35(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(cons(z0, z1))) → c35(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(fst(z0, z1))) → c35(TOP(fst(proper(z0), proper(z1))), PROPER(fst(z0, z1)))
TOP(mark(from(z0))) → c35(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(add(z0, z1))) → c35(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
TOP(mark(len(z0))) → c35(TOP(len(proper(z0))), PROPER(len(z0)))
TOP(mark(0)) → c35(TOP(ok(0)))
TOP(mark(nil)) → c35(TOP(ok(nil)))
TOP(ok(fst(0, z0))) → c36(TOP(mark(nil)), ACTIVE(fst(0, z0)))
TOP(ok(fst(s(z0), cons(z1, z2)))) → c36(TOP(mark(cons(z1, fst(z0, z2)))), ACTIVE(fst(s(z0), cons(z1, z2))))
TOP(ok(from(z0))) → c36(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(add(0, z0))) → c36(TOP(mark(z0)), ACTIVE(add(0, z0)))
TOP(ok(add(s(z0), z1))) → c36(TOP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
TOP(ok(len(nil))) → c36(TOP(mark(0)), ACTIVE(len(nil)))
TOP(ok(len(cons(z0, z1)))) → c36(TOP(mark(s(len(z1)))), ACTIVE(len(cons(z0, z1))))
TOP(ok(cons(z0, z1))) → c36(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(fst(z0, z1))) → c36(TOP(fst(active(z0), z1)), ACTIVE(fst(z0, z1)))
TOP(ok(fst(z0, z1))) → c36(TOP(fst(z0, active(z1))), ACTIVE(fst(z0, z1)))
TOP(ok(from(z0))) → c36(TOP(from(active(z0))), ACTIVE(from(z0)))
TOP(ok(add(z0, z1))) → c36(TOP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
TOP(ok(add(z0, z1))) → c36(TOP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
TOP(ok(len(z0))) → c36(TOP(len(active(z0))), ACTIVE(len(z0)))
TOP(ok(x0)) → c36

S tuples:

ACTIVE(fst(s(z0), cons(z1, z2))) → c1(CONS(z1, fst(z0, z2)), FST(z0, z2))
ACTIVE(len(cons(z0, z1))) → c6(S(len(z1)), LEN(z1))
ACTIVE(cons(z0, z1)) → c7(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c8(FST(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c9(FST(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c10(FROM(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c12(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(len(z0)) → c13(LEN(active(z0)), ACTIVE(z0))
CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
FST(mark(z0), z1) → c16(FST(z0, z1))
FST(z0, mark(z1)) → c17(FST(z0, z1))
FST(ok(z0), ok(z1)) → c18(FST(z0, z1))
FROM(mark(z0)) → c19(FROM(z0))
FROM(ok(z0)) → c20(FROM(z0))
ADD(mark(z0), z1) → c21(ADD(z0, z1))
ADD(z0, mark(z1)) → c22(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c23(ADD(z0, z1))
LEN(mark(z0)) → c24(LEN(z0))
LEN(ok(z0)) → c25(LEN(z0))
PROPER(s(z0)) → c27(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c29(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fst(z0, z1)) → c30(FST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c32(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(len(z0)) → c33(LEN(proper(z0)), PROPER(z0))
S(ok(z0)) → c34(S(z0))
ACTIVE(from(z0)) → c2(CONS(z0, from(s(z0))), S(z0))
ACTIVE(add(s(x0), x1)) → c4(ADD(x0, x1))
ACTIVE(add(s(mark(z0)), z1)) → c4(ADD(mark(z0), z1))
TOP(ok(fst(0, z0))) → c36(TOP(mark(nil)), ACTIVE(fst(0, z0)))
TOP(ok(fst(s(z0), cons(z1, z2)))) → c36(TOP(mark(cons(z1, fst(z0, z2)))), ACTIVE(fst(s(z0), cons(z1, z2))))
TOP(ok(from(z0))) → c36(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(add(0, z0))) → c36(TOP(mark(z0)), ACTIVE(add(0, z0)))
TOP(ok(add(s(z0), z1))) → c36(TOP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
TOP(ok(len(nil))) → c36(TOP(mark(0)), ACTIVE(len(nil)))
TOP(ok(len(cons(z0, z1)))) → c36(TOP(mark(s(len(z1)))), ACTIVE(len(cons(z0, z1))))
TOP(ok(cons(z0, z1))) → c36(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(fst(z0, z1))) → c36(TOP(fst(active(z0), z1)), ACTIVE(fst(z0, z1)))
TOP(ok(fst(z0, z1))) → c36(TOP(fst(z0, active(z1))), ACTIVE(fst(z0, z1)))
TOP(ok(from(z0))) → c36(TOP(from(active(z0))), ACTIVE(from(z0)))
TOP(ok(add(z0, z1))) → c36(TOP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
TOP(ok(add(z0, z1))) → c36(TOP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
TOP(ok(len(z0))) → c36(TOP(len(active(z0))), ACTIVE(len(z0)))
TOP(ok(x0)) → c36

K tuples:none
Defined Rule Symbols:

active, cons, fst, from, add, len, proper, s, top

Defined Pair Symbols:

CONS, FST, FROM, ADD, LEN, S, ACTIVE, PROPER, TOP

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c34, c1, c6, c7, c8, c9, c10, c11, c12, c13, c2, c4, c27, c29, c30, c31, c32, c33, c35, c35, c36, c36

(21) CdtUnreachableProof (EQUIVALENT transformation)

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

ACTIVE(fst(s(z0), cons(z1, z2))) → c1(CONS(z1, fst(z0, z2)), FST(z0, z2))
ACTIVE(len(cons(z0, z1))) → c6(S(len(z1)), LEN(z1))
ACTIVE(cons(z0, z1)) → c7(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c8(FST(active(z0), z1), ACTIVE(z0))
ACTIVE(fst(z0, z1)) → c9(FST(z0, active(z1)), ACTIVE(z1))
ACTIVE(from(z0)) → c10(FROM(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c11(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c12(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(len(z0)) → c13(LEN(active(z0)), ACTIVE(z0))
ACTIVE(from(z0)) → c2(CONS(z0, from(s(z0))), S(z0))
ACTIVE(add(s(x0), x1)) → c4(ADD(x0, x1))
PROPER(s(z0)) → c27(S(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c29(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(fst(z0, z1)) → c30(FST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(from(z0)) → c31(FROM(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c32(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(len(z0)) → c33(LEN(proper(z0)), PROPER(z0))
ACTIVE(add(s(mark(z0)), z1)) → c4(ADD(mark(z0), z1))
TOP(mark(s(z0))) → c35(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(cons(z0, z1))) → c35(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(fst(z0, z1))) → c35(TOP(fst(proper(z0), proper(z1))), PROPER(fst(z0, z1)))
TOP(mark(from(z0))) → c35(TOP(from(proper(z0))), PROPER(from(z0)))
TOP(mark(add(z0, z1))) → c35(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
TOP(mark(len(z0))) → c35(TOP(len(proper(z0))), PROPER(len(z0)))
TOP(ok(fst(0, z0))) → c36(TOP(mark(nil)), ACTIVE(fst(0, z0)))
TOP(ok(fst(s(z0), cons(z1, z2)))) → c36(TOP(mark(cons(z1, fst(z0, z2)))), ACTIVE(fst(s(z0), cons(z1, z2))))
TOP(ok(from(z0))) → c36(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0)))
TOP(ok(add(0, z0))) → c36(TOP(mark(z0)), ACTIVE(add(0, z0)))
TOP(ok(add(s(z0), z1))) → c36(TOP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
TOP(ok(len(nil))) → c36(TOP(mark(0)), ACTIVE(len(nil)))
TOP(ok(len(cons(z0, z1)))) → c36(TOP(mark(s(len(z1)))), ACTIVE(len(cons(z0, z1))))
TOP(ok(cons(z0, z1))) → c36(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(fst(z0, z1))) → c36(TOP(fst(active(z0), z1)), ACTIVE(fst(z0, z1)))
TOP(ok(fst(z0, z1))) → c36(TOP(fst(z0, active(z1))), ACTIVE(fst(z0, z1)))
TOP(ok(from(z0))) → c36(TOP(from(active(z0))), ACTIVE(from(z0)))
TOP(ok(add(z0, z1))) → c36(TOP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
TOP(ok(add(z0, z1))) → c36(TOP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
TOP(ok(len(z0))) → c36(TOP(len(active(z0))), ACTIVE(len(z0)))

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
FST(mark(z0), z1) → c16(FST(z0, z1))
FST(z0, mark(z1)) → c17(FST(z0, z1))
FST(ok(z0), ok(z1)) → c18(FST(z0, z1))
FROM(mark(z0)) → c19(FROM(z0))
FROM(ok(z0)) → c20(FROM(z0))
ADD(mark(z0), z1) → c21(ADD(z0, z1))
ADD(z0, mark(z1)) → c22(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c23(ADD(z0, z1))
LEN(mark(z0)) → c24(LEN(z0))
LEN(ok(z0)) → c25(LEN(z0))
S(ok(z0)) → c34(S(z0))
TOP(mark(0)) → c35(TOP(ok(0)))
TOP(mark(nil)) → c35(TOP(ok(nil)))
TOP(ok(x0)) → c36

S tuples:

CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
FST(mark(z0), z1) → c16(FST(z0, z1))
FST(z0, mark(z1)) → c17(FST(z0, z1))
FST(ok(z0), ok(z1)) → c18(FST(z0, z1))
FROM(mark(z0)) → c19(FROM(z0))
FROM(ok(z0)) → c20(FROM(z0))
ADD(mark(z0), z1) → c21(ADD(z0, z1))
ADD(z0, mark(z1)) → c22(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c23(ADD(z0, z1))
LEN(mark(z0)) → c24(LEN(z0))
LEN(ok(z0)) → c25(LEN(z0))
S(ok(z0)) → c34(S(z0))
TOP(ok(x0)) → c36

K tuples:none
Defined Rule Symbols:

active, cons, fst, from, add, len, proper, s, top

Defined Pair Symbols:

CONS, FST, FROM, ADD, LEN, S, TOP

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c34, c35, c36

(23) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 3 leading nodes:

TOP(mark(0)) → c35(TOP(ok(0)))
TOP(mark(nil)) → c35(TOP(ok(nil)))
TOP(ok(x0)) → c36

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
FST(mark(z0), z1) → c16(FST(z0, z1))
FST(z0, mark(z1)) → c17(FST(z0, z1))
FST(ok(z0), ok(z1)) → c18(FST(z0, z1))
FROM(mark(z0)) → c19(FROM(z0))
FROM(ok(z0)) → c20(FROM(z0))
ADD(mark(z0), z1) → c21(ADD(z0, z1))
ADD(z0, mark(z1)) → c22(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c23(ADD(z0, z1))
LEN(mark(z0)) → c24(LEN(z0))
LEN(ok(z0)) → c25(LEN(z0))
S(ok(z0)) → c34(S(z0))

S tuples:

CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
FST(mark(z0), z1) → c16(FST(z0, z1))
FST(z0, mark(z1)) → c17(FST(z0, z1))
FST(ok(z0), ok(z1)) → c18(FST(z0, z1))
FROM(mark(z0)) → c19(FROM(z0))
FROM(ok(z0)) → c20(FROM(z0))
ADD(mark(z0), z1) → c21(ADD(z0, z1))
ADD(z0, mark(z1)) → c22(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c23(ADD(z0, z1))
LEN(mark(z0)) → c24(LEN(z0))
LEN(ok(z0)) → c25(LEN(z0))
S(ok(z0)) → c34(S(z0))

K tuples:none
Defined Rule Symbols:

active, cons, fst, from, add, len, proper, s, top

Defined Pair Symbols:

CONS, FST, FROM, ADD, LEN, S

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c34

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

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

CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
FST(mark(z0), z1) → c16(FST(z0, z1))
FST(z0, mark(z1)) → c17(FST(z0, z1))
FST(ok(z0), ok(z1)) → c18(FST(z0, z1))
FROM(mark(z0)) → c19(FROM(z0))
FROM(ok(z0)) → c20(FROM(z0))
ADD(mark(z0), z1) → c21(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c23(ADD(z0, z1))
S(ok(z0)) → c34(S(z0))

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

CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
FST(mark(z0), z1) → c16(FST(z0, z1))
FST(z0, mark(z1)) → c17(FST(z0, z1))
FST(ok(z0), ok(z1)) → c18(FST(z0, z1))
FROM(mark(z0)) → c19(FROM(z0))
FROM(ok(z0)) → c20(FROM(z0))
ADD(mark(z0), z1) → c21(ADD(z0, z1))
ADD(z0, mark(z1)) → c22(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c23(ADD(z0, z1))
LEN(mark(z0)) → c24(LEN(z0))
LEN(ok(z0)) → c25(LEN(z0))
S(ok(z0)) → c34(S(z0))

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

POL(ADD(x₁, x₂)) = x₁
POL(CONS(x₁, x₂)) = x₁
POL(FROM(x₁)) = x₁
POL(FST(x₁, x₂)) = x₁ + x₂
POL(LEN(x₁)) = 0
POL(S(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)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = [2] + x₁

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
FST(mark(z0), z1) → c16(FST(z0, z1))
FST(z0, mark(z1)) → c17(FST(z0, z1))
FST(ok(z0), ok(z1)) → c18(FST(z0, z1))
FROM(mark(z0)) → c19(FROM(z0))
FROM(ok(z0)) → c20(FROM(z0))
ADD(mark(z0), z1) → c21(ADD(z0, z1))
ADD(z0, mark(z1)) → c22(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c23(ADD(z0, z1))
LEN(mark(z0)) → c24(LEN(z0))
LEN(ok(z0)) → c25(LEN(z0))
S(ok(z0)) → c34(S(z0))

S tuples:

ADD(z0, mark(z1)) → c22(ADD(z0, z1))
LEN(mark(z0)) → c24(LEN(z0))
LEN(ok(z0)) → c25(LEN(z0))

K tuples:

CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
FST(mark(z0), z1) → c16(FST(z0, z1))
FST(z0, mark(z1)) → c17(FST(z0, z1))
FST(ok(z0), ok(z1)) → c18(FST(z0, z1))
FROM(mark(z0)) → c19(FROM(z0))
FROM(ok(z0)) → c20(FROM(z0))
ADD(mark(z0), z1) → c21(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c23(ADD(z0, z1))
S(ok(z0)) → c34(S(z0))

Defined Rule Symbols:

active, cons, fst, from, add, len, proper, s, top

Defined Pair Symbols:

CONS, FST, FROM, ADD, LEN, S

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c34

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

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

LEN(mark(z0)) → c24(LEN(z0))

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

CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
FST(mark(z0), z1) → c16(FST(z0, z1))
FST(z0, mark(z1)) → c17(FST(z0, z1))
FST(ok(z0), ok(z1)) → c18(FST(z0, z1))
FROM(mark(z0)) → c19(FROM(z0))
FROM(ok(z0)) → c20(FROM(z0))
ADD(mark(z0), z1) → c21(ADD(z0, z1))
ADD(z0, mark(z1)) → c22(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c23(ADD(z0, z1))
LEN(mark(z0)) → c24(LEN(z0))
LEN(ok(z0)) → c25(LEN(z0))
S(ok(z0)) → c34(S(z0))

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

POL(ADD(x₁, x₂)) = 0
POL(CONS(x₁, x₂)) = 0
POL(FROM(x₁)) = [2]x₁
POL(FST(x₁, x₂)) = 0
POL(LEN(x₁)) = x₁
POL(S(x₁)) = [2]x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(mark(x₁)) = [2] + x₁
POL(ok(x₁)) = x₁

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
FST(mark(z0), z1) → c16(FST(z0, z1))
FST(z0, mark(z1)) → c17(FST(z0, z1))
FST(ok(z0), ok(z1)) → c18(FST(z0, z1))
FROM(mark(z0)) → c19(FROM(z0))
FROM(ok(z0)) → c20(FROM(z0))
ADD(mark(z0), z1) → c21(ADD(z0, z1))
ADD(z0, mark(z1)) → c22(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c23(ADD(z0, z1))
LEN(mark(z0)) → c24(LEN(z0))
LEN(ok(z0)) → c25(LEN(z0))
S(ok(z0)) → c34(S(z0))

S tuples:

ADD(z0, mark(z1)) → c22(ADD(z0, z1))
LEN(ok(z0)) → c25(LEN(z0))

K tuples:

CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
FST(mark(z0), z1) → c16(FST(z0, z1))
FST(z0, mark(z1)) → c17(FST(z0, z1))
FST(ok(z0), ok(z1)) → c18(FST(z0, z1))
FROM(mark(z0)) → c19(FROM(z0))
FROM(ok(z0)) → c20(FROM(z0))
ADD(mark(z0), z1) → c21(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c23(ADD(z0, z1))
S(ok(z0)) → c34(S(z0))
LEN(mark(z0)) → c24(LEN(z0))

Defined Rule Symbols:

active, cons, fst, from, add, len, proper, s, top

Defined Pair Symbols:

CONS, FST, FROM, ADD, LEN, S

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c34

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

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

ADD(z0, mark(z1)) → c22(ADD(z0, z1))

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

CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
FST(mark(z0), z1) → c16(FST(z0, z1))
FST(z0, mark(z1)) → c17(FST(z0, z1))
FST(ok(z0), ok(z1)) → c18(FST(z0, z1))
FROM(mark(z0)) → c19(FROM(z0))
FROM(ok(z0)) → c20(FROM(z0))
ADD(mark(z0), z1) → c21(ADD(z0, z1))
ADD(z0, mark(z1)) → c22(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c23(ADD(z0, z1))
LEN(mark(z0)) → c24(LEN(z0))
LEN(ok(z0)) → c25(LEN(z0))
S(ok(z0)) → c34(S(z0))

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

POL(ADD(x₁, x₂)) = x₂
POL(CONS(x₁, x₂)) = [4]x₁ + [4]x₂
POL(FROM(x₁)) = x₁
POL(FST(x₁, x₂)) = 0
POL(LEN(x₁)) = 0
POL(S(x₁)) = [2]x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
FST(mark(z0), z1) → c16(FST(z0, z1))
FST(z0, mark(z1)) → c17(FST(z0, z1))
FST(ok(z0), ok(z1)) → c18(FST(z0, z1))
FROM(mark(z0)) → c19(FROM(z0))
FROM(ok(z0)) → c20(FROM(z0))
ADD(mark(z0), z1) → c21(ADD(z0, z1))
ADD(z0, mark(z1)) → c22(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c23(ADD(z0, z1))
LEN(mark(z0)) → c24(LEN(z0))
LEN(ok(z0)) → c25(LEN(z0))
S(ok(z0)) → c34(S(z0))

S tuples:

LEN(ok(z0)) → c25(LEN(z0))

K tuples:

CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
FST(mark(z0), z1) → c16(FST(z0, z1))
FST(z0, mark(z1)) → c17(FST(z0, z1))
FST(ok(z0), ok(z1)) → c18(FST(z0, z1))
FROM(mark(z0)) → c19(FROM(z0))
FROM(ok(z0)) → c20(FROM(z0))
ADD(mark(z0), z1) → c21(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c23(ADD(z0, z1))
S(ok(z0)) → c34(S(z0))
LEN(mark(z0)) → c24(LEN(z0))
ADD(z0, mark(z1)) → c22(ADD(z0, z1))

Defined Rule Symbols:

active, cons, fst, from, add, len, proper, s, top

Defined Pair Symbols:

CONS, FST, FROM, ADD, LEN, S

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c34

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

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

LEN(ok(z0)) → c25(LEN(z0))

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

CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
FST(mark(z0), z1) → c16(FST(z0, z1))
FST(z0, mark(z1)) → c17(FST(z0, z1))
FST(ok(z0), ok(z1)) → c18(FST(z0, z1))
FROM(mark(z0)) → c19(FROM(z0))
FROM(ok(z0)) → c20(FROM(z0))
ADD(mark(z0), z1) → c21(ADD(z0, z1))
ADD(z0, mark(z1)) → c22(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c23(ADD(z0, z1))
LEN(mark(z0)) → c24(LEN(z0))
LEN(ok(z0)) → c25(LEN(z0))
S(ok(z0)) → c34(S(z0))

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

POL(ADD(x₁, x₂)) = [4]x₁
POL(CONS(x₁, x₂)) = 0
POL(FROM(x₁)) = [3]x₁
POL(FST(x₁, x₂)) = [3]x₁ + [2]x₂
POL(LEN(x₁)) = [4]x₁
POL(S(x₁)) = [4]x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(ok(x₁)) = [1] + x₁

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
FST(mark(z0), z1) → c16(FST(z0, z1))
FST(z0, mark(z1)) → c17(FST(z0, z1))
FST(ok(z0), ok(z1)) → c18(FST(z0, z1))
FROM(mark(z0)) → c19(FROM(z0))
FROM(ok(z0)) → c20(FROM(z0))
ADD(mark(z0), z1) → c21(ADD(z0, z1))
ADD(z0, mark(z1)) → c22(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c23(ADD(z0, z1))
LEN(mark(z0)) → c24(LEN(z0))
LEN(ok(z0)) → c25(LEN(z0))
S(ok(z0)) → c34(S(z0))

S tuples:none
K tuples:

CONS(mark(z0), z1) → c14(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c15(CONS(z0, z1))
FST(mark(z0), z1) → c16(FST(z0, z1))
FST(z0, mark(z1)) → c17(FST(z0, z1))
FST(ok(z0), ok(z1)) → c18(FST(z0, z1))
FROM(mark(z0)) → c19(FROM(z0))
FROM(ok(z0)) → c20(FROM(z0))
ADD(mark(z0), z1) → c21(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c23(ADD(z0, z1))
S(ok(z0)) → c34(S(z0))
LEN(mark(z0)) → c24(LEN(z0))
ADD(z0, mark(z1)) → c22(ADD(z0, z1))
LEN(ok(z0)) → c25(LEN(z0))

Defined Rule Symbols:

active, cons, fst, from, add, len, proper, s, top

Defined Pair Symbols:

CONS, FST, FROM, ADD, LEN, S

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c34

(33) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(0) Obligation:

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

(2) Obligation:

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

(4) Obligation:

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

(6) Obligation:

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

(8) Obligation:

(9) CdtUnreachableProof (EQUIVALENT transformation)

(10) Obligation:

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

(12) Obligation:

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

(14) Obligation:

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

(16) Obligation:

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

(18) Obligation:

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

(20) Obligation:

(21) CdtUnreachableProof (EQUIVALENT transformation)

(22) Obligation:

(23) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

(24) Obligation:

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

(26) Obligation:

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

(28) Obligation:

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

(30) Obligation:

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

(32) Obligation:

(33) SIsEmptyProof (EQUIVALENT transformation)

(34) BOUNDS(O(1), O(1))