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

0 CpxTRS
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)
↳2 CdtProblem
↳3 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳4 CdtProblem
↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳6 CdtProblem
↳7 CdtUsableRulesProof (⇔, 10 ms)
↳8 CdtProblem
↳9 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 27 ms)
↳10 CdtProblem
↳11 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳12 CdtProblem
↳13 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳14 CdtProblem
↳15 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳16 CdtProblem
↳17 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 2 ms)
↳18 CdtProblem
↳19 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳20 CdtProblem
↳21 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳22 CdtProblem
↳23 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳24 CdtProblem
↳25 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳26 CdtProblem
↳27 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳28 CdtProblem
↳29 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳30 CdtProblem
↳31 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳32 CdtProblem
↳33 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 6 ms)
↳34 CdtProblem
↳35 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳36 CdtProblem
↳37 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳38 CdtProblem
↳39 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 4 ms)
↳40 CdtProblem
↳41 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳42 CdtProblem
↳43 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳44 CdtProblem
↳45 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 4256 ms)
↳46 CdtProblem
↳47 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳48 CdtProblem
↳49 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳50 CdtProblem
↳51 CdtUnreachableProof (⇔, 0 ms)
↳52 CdtProblem
↳53 CdtUsableRulesProof (⇔, 0 ms)
↳54 CdtProblem
↳55 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 0 ms)
↳56 CdtProblem
↳57 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 5 ms)
↳58 CdtProblem
↳59 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 9 ms)
↳60 CdtProblem
↳61 CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)), 0 ms)
↳62 CdtProblem
↳63 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)
↳64 BOUNDS(1, 1)

(0) Obligation:

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

active(f(X)) → mark(cons(X, f(g(X))))
active(g(0)) → mark(s(0))
active(g(s(X))) → mark(s(s(g(X))))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(g(X)) → g(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
g(mark(X)) → mark(g(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(f(X)) → f(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(g(X)) → g(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
f(ok(X)) → ok(f(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
g(ok(X)) → ok(g(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
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(f(z0)) → mark(cons(z0, f(g(z0))))
active(g(0)) → mark(s(0))
active(g(s(z0))) → mark(s(s(g(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(f(z0)) → f(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(g(z0)) → g(active(z0))
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
proper(f(z0)) → f(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(g(z0)) → g(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0)) → c(CONS(z0, f(g(z0))), F(g(z0)), G(z0))
ACTIVE(g(0)) → c1(S(0))
ACTIVE(g(s(z0))) → c2(S(s(g(z0))), S(g(z0)), G(z0))
ACTIVE(sel(0, cons(z0, z1))) → c3
ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(f(z0)) → c5(F(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(g(z0)) → c7(G(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0))
PROPER(0) → c25
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(f(z0)) → c(CONS(z0, f(g(z0))), F(g(z0)), G(z0))
ACTIVE(g(0)) → c1(S(0))
ACTIVE(g(s(z0))) → c2(S(s(g(z0))), S(g(z0)), G(z0))
ACTIVE(sel(0, cons(z0, z1))) → c3
ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(f(z0)) → c5(F(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(g(z0)) → c7(G(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0))
PROPER(0) → c25
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:

active, f, cons, g, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, F, CONS, G, S, SEL, PROPER, TOP

Compound Symbols:

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

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

Removed 3 trailing nodes:

ACTIVE(sel(0, cons(z0, z1))) → c3
ACTIVE(g(0)) → c1(S(0))
PROPER(0) → c25

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0))))
active(g(0)) → mark(s(0))
active(g(s(z0))) → mark(s(s(g(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(f(z0)) → f(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(g(z0)) → g(active(z0))
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
proper(f(z0)) → f(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(g(z0)) → g(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0)) → c(CONS(z0, f(g(z0))), F(g(z0)), G(z0))
ACTIVE(g(s(z0))) → c2(S(s(g(z0))), S(g(z0)), G(z0))
ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(f(z0)) → c5(F(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(g(z0)) → c7(G(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(f(z0)) → c(CONS(z0, f(g(z0))), F(g(z0)), G(z0))
ACTIVE(g(s(z0))) → c2(S(s(g(z0))), S(g(z0)), G(z0))
ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(f(z0)) → c5(F(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(g(z0)) → c7(G(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:

active, f, cons, g, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, F, CONS, G, S, SEL, PROPER, TOP

Compound Symbols:

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

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

Removed 4 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0))))
active(g(0)) → mark(s(0))
active(g(s(z0))) → mark(s(s(g(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(f(z0)) → f(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(g(z0)) → g(active(z0))
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
proper(f(z0)) → f(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(g(z0)) → g(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(f(z0)) → c5(F(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(g(z0)) → c7(G(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(f(z0)) → c5(F(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(g(z0)) → c7(G(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
K tuples:none
Defined Rule Symbols:

active, f, cons, g, s, sel, proper, top

Defined Pair Symbols:

ACTIVE, F, CONS, G, S, SEL, PROPER, TOP

Compound Symbols:

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

(7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

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

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0))))
active(g(0)) → mark(s(0))
active(g(s(z0))) → mark(s(s(g(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(f(z0)) → f(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(g(z0)) → g(active(z0))
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(f(z0)) → f(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(g(z0)) → g(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(f(z0)) → c5(F(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(g(z0)) → c7(G(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(f(z0)) → c5(F(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(g(z0)) → c7(G(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
K tuples:none
Defined Rule Symbols:

active, sel, f, cons, g, s, proper

Defined Pair Symbols:

ACTIVE, F, CONS, G, S, SEL, PROPER, TOP

Compound Symbols:

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

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

Use narrowing to replace ACTIVE(f(z0)) → c5(F(active(z0)), ACTIVE(z0)) by

ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))), ACTIVE(g(0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0))))
active(g(0)) → mark(s(0))
active(g(s(z0))) → mark(s(s(g(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(f(z0)) → f(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(g(z0)) → g(active(z0))
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(f(z0)) → f(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(g(z0)) → g(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(g(z0)) → c7(G(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))), ACTIVE(g(0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(g(z0)) → c7(G(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))), ACTIVE(g(0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, sel, f, cons, g, s, proper

Defined Pair Symbols:

ACTIVE, F, CONS, G, S, SEL, PROPER, TOP

Compound Symbols:

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

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

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

ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0)))
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1), ACTIVE(g(0)))
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0))))
active(g(0)) → mark(s(0))
active(g(s(z0))) → mark(s(s(g(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(f(z0)) → f(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(g(z0)) → g(active(z0))
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(f(z0)) → f(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(g(z0)) → g(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(g(z0)) → c7(G(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))), ACTIVE(g(0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0)))
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1), ACTIVE(g(0)))
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(g(z0)) → c7(G(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))), ACTIVE(g(0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0)))
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1), ACTIVE(g(0)))
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, sel, f, cons, g, s, proper

Defined Pair Symbols:

ACTIVE, F, CONS, G, S, SEL, PROPER, TOP

Compound Symbols:

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

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

Use narrowing to replace ACTIVE(g(z0)) → c7(G(active(z0)), ACTIVE(z0)) by

ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(g(g(0))) → c7(G(mark(s(0))), ACTIVE(g(0)))
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0))))
active(g(0)) → mark(s(0))
active(g(s(z0))) → mark(s(s(g(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(f(z0)) → f(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(g(z0)) → g(active(z0))
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(f(z0)) → f(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(g(z0)) → g(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))), ACTIVE(g(0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0)))
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1), ACTIVE(g(0)))
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(g(g(0))) → c7(G(mark(s(0))), ACTIVE(g(0)))
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))), ACTIVE(g(0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0)))
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1), ACTIVE(g(0)))
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(g(g(0))) → c7(G(mark(s(0))), ACTIVE(g(0)))
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, sel, f, cons, g, s, proper

Defined Pair Symbols:

ACTIVE, F, CONS, G, S, SEL, PROPER, TOP

Compound Symbols:

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

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

Removed 3 trailing tuple parts

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0))))
active(g(0)) → mark(s(0))
active(g(s(z0))) → mark(s(s(g(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(f(z0)) → f(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(g(z0)) → g(active(z0))
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(f(z0)) → f(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(g(z0)) → g(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0)))
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))))
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1))
ACTIVE(g(g(0))) → c7(G(mark(s(0))))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0)))
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))))
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1))
ACTIVE(g(g(0))) → c7(G(mark(s(0))))
K tuples:none
Defined Rule Symbols:

active, sel, f, cons, g, s, proper

Defined Pair Symbols:

ACTIVE, F, CONS, G, S, SEL, PROPER, TOP

Compound Symbols:

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

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

Use narrowing to replace ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0)) by

ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(s(g(0))) → c8(S(mark(s(0))), ACTIVE(g(0)))
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0))))
active(g(0)) → mark(s(0))
active(g(s(z0))) → mark(s(s(g(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(f(z0)) → f(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(g(z0)) → g(active(z0))
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(f(z0)) → f(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(g(z0)) → g(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0)))
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))))
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1))
ACTIVE(g(g(0))) → c7(G(mark(s(0))))
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(s(g(0))) → c8(S(mark(s(0))), ACTIVE(g(0)))
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0)))
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))))
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1))
ACTIVE(g(g(0))) → c7(G(mark(s(0))))
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(s(g(0))) → c8(S(mark(s(0))), ACTIVE(g(0)))
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, sel, f, cons, g, s, proper

Defined Pair Symbols:

ACTIVE, F, CONS, G, S, SEL, PROPER, TOP

Compound Symbols:

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

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

Removed 1 trailing tuple parts

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0))))
active(g(0)) → mark(s(0))
active(g(s(z0))) → mark(s(s(g(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(f(z0)) → f(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(g(z0)) → g(active(z0))
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(f(z0)) → f(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(g(z0)) → g(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0)))
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))))
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1))
ACTIVE(g(g(0))) → c7(G(mark(s(0))))
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(g(0))) → c8(S(mark(s(0))))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0)))
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))))
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1))
ACTIVE(g(g(0))) → c7(G(mark(s(0))))
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(g(0))) → c8(S(mark(s(0))))
K tuples:none
Defined Rule Symbols:

active, sel, f, cons, g, s, proper

Defined Pair Symbols:

ACTIVE, F, CONS, G, S, SEL, PROPER, TOP

Compound Symbols:

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

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

Use narrowing to replace PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0)) by

PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(0)) → c22(F(ok(0)), PROPER(0))
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0)))
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0))))
active(g(0)) → mark(s(0))
active(g(s(z0))) → mark(s(s(g(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(f(z0)) → f(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(g(z0)) → g(active(z0))
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(f(z0)) → f(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(g(z0)) → g(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0)))
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))))
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1))
ACTIVE(g(g(0))) → c7(G(mark(s(0))))
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(g(0))) → c8(S(mark(s(0))))
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(0)) → c22(F(ok(0)), PROPER(0))
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0)))
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0)))
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))))
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1))
ACTIVE(g(g(0))) → c7(G(mark(s(0))))
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(g(0))) → c8(S(mark(s(0))))
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(0)) → c22(F(ok(0)), PROPER(0))
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0)))
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, sel, f, cons, g, s, proper

Defined Pair Symbols:

ACTIVE, F, CONS, G, S, SEL, PROPER, TOP

Compound Symbols:

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

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

Removed 1 trailing tuple parts

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0))))
active(g(0)) → mark(s(0))
active(g(s(z0))) → mark(s(s(g(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(f(z0)) → f(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(g(z0)) → g(active(z0))
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(f(z0)) → f(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(g(z0)) → g(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0)))
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))))
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1))
ACTIVE(g(g(0))) → c7(G(mark(s(0))))
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(g(0))) → c8(S(mark(s(0))))
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0)))
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(f(0)) → c22(F(ok(0)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0)))
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))))
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1))
ACTIVE(g(g(0))) → c7(G(mark(s(0))))
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(g(0))) → c8(S(mark(s(0))))
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0)))
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(f(0)) → c22(F(ok(0)))
K tuples:none
Defined Rule Symbols:

active, sel, f, cons, g, s, proper

Defined Pair Symbols:

ACTIVE, F, CONS, G, S, SEL, PROPER, TOP

Compound Symbols:

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

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

Use narrowing to replace PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0))))
active(g(0)) → mark(s(0))
active(g(s(z0))) → mark(s(s(g(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(f(z0)) → f(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(g(z0)) → g(active(z0))
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(f(z0)) → f(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(g(z0)) → g(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0)))
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))))
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1))
ACTIVE(g(g(0))) → c7(G(mark(s(0))))
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(g(0))) → c8(S(mark(s(0))))
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0)))
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(f(0)) → c22(F(ok(0)))
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0)))
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))))
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1))
ACTIVE(g(g(0))) → c7(G(mark(s(0))))
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(g(0))) → c8(S(mark(s(0))))
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0)))
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(f(0)) → c22(F(ok(0)))
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, sel, f, cons, g, s, proper

Defined Pair Symbols:

ACTIVE, F, CONS, G, S, SEL, PROPER, TOP

Compound Symbols:

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

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

Removed 2 trailing tuple parts

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0))))
active(g(0)) → mark(s(0))
active(g(s(z0))) → mark(s(s(g(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(f(z0)) → f(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(g(z0)) → g(active(z0))
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(f(z0)) → f(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(g(z0)) → g(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0)))
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))))
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1))
ACTIVE(g(g(0))) → c7(G(mark(s(0))))
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(g(0))) → c8(S(mark(s(0))))
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0)))
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(f(0)) → c22(F(ok(0)))
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0)))
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))))
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1))
ACTIVE(g(g(0))) → c7(G(mark(s(0))))
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(g(0))) → c8(S(mark(s(0))))
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0)))
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(f(0)) → c22(F(ok(0)))
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, sel, f, cons, g, s, proper

Defined Pair Symbols:

ACTIVE, F, CONS, G, S, SEL, PROPER, TOP

Compound Symbols:

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

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

Use narrowing to replace PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0)) by

PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(0)) → c24(G(ok(0)), PROPER(0))
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0)))
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0))))
active(g(0)) → mark(s(0))
active(g(s(z0))) → mark(s(s(g(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(f(z0)) → f(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(g(z0)) → g(active(z0))
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(f(z0)) → f(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(g(z0)) → g(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0)))
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))))
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1))
ACTIVE(g(g(0))) → c7(G(mark(s(0))))
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(g(0))) → c8(S(mark(s(0))))
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0)))
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(f(0)) → c22(F(ok(0)))
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(0)) → c24(G(ok(0)), PROPER(0))
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0)))
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0)))
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))))
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1))
ACTIVE(g(g(0))) → c7(G(mark(s(0))))
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(g(0))) → c8(S(mark(s(0))))
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0)))
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(f(0)) → c22(F(ok(0)))
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(0)) → c24(G(ok(0)), PROPER(0))
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0)))
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, sel, f, cons, g, s, proper

Defined Pair Symbols:

ACTIVE, F, CONS, G, S, SEL, PROPER, TOP

Compound Symbols:

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

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

Removed 1 trailing tuple parts

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0))))
active(g(0)) → mark(s(0))
active(g(s(z0))) → mark(s(s(g(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(f(z0)) → f(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(g(z0)) → g(active(z0))
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(f(z0)) → f(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(g(z0)) → g(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0)))
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))))
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1))
ACTIVE(g(g(0))) → c7(G(mark(s(0))))
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(g(0))) → c8(S(mark(s(0))))
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0)))
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(f(0)) → c22(F(ok(0)))
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0)))
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(g(0)) → c24(G(ok(0)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0))
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0)))
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))))
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1))
ACTIVE(g(g(0))) → c7(G(mark(s(0))))
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(g(0))) → c8(S(mark(s(0))))
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0)))
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(f(0)) → c22(F(ok(0)))
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0)))
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(g(0)) → c24(G(ok(0)))
K tuples:none
Defined Rule Symbols:

active, sel, f, cons, g, s, proper

Defined Pair Symbols:

ACTIVE, F, CONS, G, S, SEL, PROPER, TOP

Compound Symbols:

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

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

Use narrowing to replace PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) by

PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0)))
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0)))
PROPER(s(0)) → c26(S(ok(0)), PROPER(0))
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0))))
active(g(0)) → mark(s(0))
active(g(s(z0))) → mark(s(s(g(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(f(z0)) → f(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(g(z0)) → g(active(z0))
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(f(z0)) → f(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(g(z0)) → g(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0)))
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))))
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1))
ACTIVE(g(g(0))) → c7(G(mark(s(0))))
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(g(0))) → c8(S(mark(s(0))))
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0)))
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(f(0)) → c22(F(ok(0)))
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0)))
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(g(0)) → c24(G(ok(0)))
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0)))
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0)))
PROPER(s(0)) → c26(S(ok(0)), PROPER(0))
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0)))
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))))
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1))
ACTIVE(g(g(0))) → c7(G(mark(s(0))))
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(g(0))) → c8(S(mark(s(0))))
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0)))
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(f(0)) → c22(F(ok(0)))
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0)))
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(g(0)) → c24(G(ok(0)))
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0)))
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0)))
PROPER(s(0)) → c26(S(ok(0)), PROPER(0))
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, sel, f, cons, g, s, proper

Defined Pair Symbols:

ACTIVE, F, CONS, G, S, SEL, PROPER, TOP

Compound Symbols:

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

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

Removed 1 trailing tuple parts

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0))))
active(g(0)) → mark(s(0))
active(g(s(z0))) → mark(s(s(g(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(f(z0)) → f(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(g(z0)) → g(active(z0))
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(f(z0)) → f(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(g(z0)) → g(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0)))
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))))
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1))
ACTIVE(g(g(0))) → c7(G(mark(s(0))))
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(g(0))) → c8(S(mark(s(0))))
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0)))
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(f(0)) → c22(F(ok(0)))
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0)))
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(g(0)) → c24(G(ok(0)))
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0)))
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0)))
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c26(S(ok(0)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0)))
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))))
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1))
ACTIVE(g(g(0))) → c7(G(mark(s(0))))
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(g(0))) → c8(S(mark(s(0))))
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0)))
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(f(0)) → c22(F(ok(0)))
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0)))
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(g(0)) → c24(G(ok(0)))
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0)))
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0)))
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c26(S(ok(0)))
K tuples:none
Defined Rule Symbols:

active, sel, f, cons, g, s, proper

Defined Pair Symbols:

ACTIVE, F, CONS, G, S, SEL, PROPER, TOP

Compound Symbols:

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

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

Use narrowing to replace PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(sel(x0, f(z0))) → c27(SEL(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(sel(x0, cons(z0, z1))) → c27(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, g(z0))) → c27(SEL(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(sel(x0, 0)) → c27(SEL(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(sel(x0, s(z0))) → c27(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, sel(z0, z1))) → c27(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(f(z0), x1)) → c27(SEL(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c27(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(g(z0), x1)) → c27(SEL(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(sel(0, x1)) → c27(SEL(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(sel(s(z0), x1)) → c27(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c27(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0))))
active(g(0)) → mark(s(0))
active(g(s(z0))) → mark(s(s(g(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(f(z0)) → f(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(g(z0)) → g(active(z0))
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(f(z0)) → f(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(g(z0)) → g(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0)))
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))))
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1))
ACTIVE(g(g(0))) → c7(G(mark(s(0))))
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(g(0))) → c8(S(mark(s(0))))
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0)))
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(f(0)) → c22(F(ok(0)))
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0)))
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(g(0)) → c24(G(ok(0)))
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0)))
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0)))
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c26(S(ok(0)))
PROPER(sel(x0, f(z0))) → c27(SEL(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(sel(x0, cons(z0, z1))) → c27(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, g(z0))) → c27(SEL(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(sel(x0, 0)) → c27(SEL(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(sel(x0, s(z0))) → c27(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, sel(z0, z1))) → c27(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(f(z0), x1)) → c27(SEL(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c27(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(g(z0), x1)) → c27(SEL(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(sel(0, x1)) → c27(SEL(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(sel(s(z0), x1)) → c27(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c27(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0)))
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))))
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1))
ACTIVE(g(g(0))) → c7(G(mark(s(0))))
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(g(0))) → c8(S(mark(s(0))))
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0)))
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(f(0)) → c22(F(ok(0)))
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0)))
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(g(0)) → c24(G(ok(0)))
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0)))
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0)))
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c26(S(ok(0)))
PROPER(sel(x0, f(z0))) → c27(SEL(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(sel(x0, cons(z0, z1))) → c27(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, g(z0))) → c27(SEL(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(sel(x0, 0)) → c27(SEL(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(sel(x0, s(z0))) → c27(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, sel(z0, z1))) → c27(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(f(z0), x1)) → c27(SEL(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c27(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(g(z0), x1)) → c27(SEL(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(sel(0, x1)) → c27(SEL(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(sel(s(z0), x1)) → c27(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c27(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, sel, f, cons, g, s, proper

Defined Pair Symbols:

ACTIVE, F, CONS, G, S, SEL, TOP, PROPER

Compound Symbols:

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

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

Removed 2 trailing tuple parts

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0))))
active(g(0)) → mark(s(0))
active(g(s(z0))) → mark(s(s(g(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(f(z0)) → f(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(g(z0)) → g(active(z0))
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(f(z0)) → f(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(g(z0)) → g(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0)))
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))))
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1))
ACTIVE(g(g(0))) → c7(G(mark(s(0))))
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(g(0))) → c8(S(mark(s(0))))
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0)))
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(f(0)) → c22(F(ok(0)))
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0)))
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(g(0)) → c24(G(ok(0)))
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0)))
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0)))
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c26(S(ok(0)))
PROPER(sel(x0, f(z0))) → c27(SEL(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(sel(x0, cons(z0, z1))) → c27(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, g(z0))) → c27(SEL(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(sel(x0, s(z0))) → c27(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, sel(z0, z1))) → c27(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(f(z0), x1)) → c27(SEL(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c27(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(g(z0), x1)) → c27(SEL(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c27(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c27(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c27(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c27(SEL(ok(0), proper(x1)), PROPER(x1))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0)))
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))))
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1))
ACTIVE(g(g(0))) → c7(G(mark(s(0))))
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(g(0))) → c8(S(mark(s(0))))
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0)))
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(f(0)) → c22(F(ok(0)))
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0)))
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(g(0)) → c24(G(ok(0)))
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0)))
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0)))
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c26(S(ok(0)))
PROPER(sel(x0, f(z0))) → c27(SEL(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(sel(x0, cons(z0, z1))) → c27(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, g(z0))) → c27(SEL(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(sel(x0, s(z0))) → c27(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, sel(z0, z1))) → c27(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(f(z0), x1)) → c27(SEL(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c27(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(g(z0), x1)) → c27(SEL(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c27(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c27(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c27(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c27(SEL(ok(0), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, sel, f, cons, g, s, proper

Defined Pair Symbols:

ACTIVE, F, CONS, G, S, SEL, TOP, PROPER

Compound Symbols:

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

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

Use narrowing to replace TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(f(z0))) → c28(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(cons(z0, z1))) → c28(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(g(z0))) → c28(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(0)) → c28(TOP(ok(0)), PROPER(0))
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(sel(z0, z1))) → c28(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0))))
active(g(0)) → mark(s(0))
active(g(s(z0))) → mark(s(s(g(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(f(z0)) → f(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(g(z0)) → g(active(z0))
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(f(z0)) → f(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(g(z0)) → g(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0)))
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))))
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1))
ACTIVE(g(g(0))) → c7(G(mark(s(0))))
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(g(0))) → c8(S(mark(s(0))))
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0)))
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(f(0)) → c22(F(ok(0)))
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0)))
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(g(0)) → c24(G(ok(0)))
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0)))
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0)))
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c26(S(ok(0)))
PROPER(sel(x0, f(z0))) → c27(SEL(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(sel(x0, cons(z0, z1))) → c27(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, g(z0))) → c27(SEL(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(sel(x0, s(z0))) → c27(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, sel(z0, z1))) → c27(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(f(z0), x1)) → c27(SEL(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c27(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(g(z0), x1)) → c27(SEL(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c27(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c27(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c27(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c27(SEL(ok(0), proper(x1)), PROPER(x1))
TOP(mark(f(z0))) → c28(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(cons(z0, z1))) → c28(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(g(z0))) → c28(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(0)) → c28(TOP(ok(0)), PROPER(0))
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(sel(z0, z1))) → c28(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0)))
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))))
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1))
ACTIVE(g(g(0))) → c7(G(mark(s(0))))
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(g(0))) → c8(S(mark(s(0))))
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0)))
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(f(0)) → c22(F(ok(0)))
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0)))
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(g(0)) → c24(G(ok(0)))
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0)))
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0)))
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c26(S(ok(0)))
PROPER(sel(x0, f(z0))) → c27(SEL(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(sel(x0, cons(z0, z1))) → c27(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, g(z0))) → c27(SEL(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(sel(x0, s(z0))) → c27(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, sel(z0, z1))) → c27(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(f(z0), x1)) → c27(SEL(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c27(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(g(z0), x1)) → c27(SEL(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c27(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c27(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c27(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c27(SEL(ok(0), proper(x1)), PROPER(x1))
TOP(mark(f(z0))) → c28(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(cons(z0, z1))) → c28(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(g(z0))) → c28(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(0)) → c28(TOP(ok(0)), PROPER(0))
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(sel(z0, z1))) → c28(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, sel, f, cons, g, s, proper

Defined Pair Symbols:

ACTIVE, F, CONS, G, S, SEL, TOP, PROPER

Compound Symbols:

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

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

Removed 1 trailing tuple parts

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0))))
active(g(0)) → mark(s(0))
active(g(s(z0))) → mark(s(s(g(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(f(z0)) → f(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(g(z0)) → g(active(z0))
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(f(z0)) → f(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(g(z0)) → g(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0)))
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))))
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1))
ACTIVE(g(g(0))) → c7(G(mark(s(0))))
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(g(0))) → c8(S(mark(s(0))))
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0)))
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(f(0)) → c22(F(ok(0)))
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0)))
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(g(0)) → c24(G(ok(0)))
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0)))
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0)))
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c26(S(ok(0)))
PROPER(sel(x0, f(z0))) → c27(SEL(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(sel(x0, cons(z0, z1))) → c27(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, g(z0))) → c27(SEL(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(sel(x0, s(z0))) → c27(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, sel(z0, z1))) → c27(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(f(z0), x1)) → c27(SEL(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c27(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(g(z0), x1)) → c27(SEL(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c27(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c27(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c27(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c27(SEL(ok(0), proper(x1)), PROPER(x1))
TOP(mark(f(z0))) → c28(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(cons(z0, z1))) → c28(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(g(z0))) → c28(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(sel(z0, z1))) → c28(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(0)) → c28(TOP(ok(0)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0)))
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))))
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1))
ACTIVE(g(g(0))) → c7(G(mark(s(0))))
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(g(0))) → c8(S(mark(s(0))))
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0)))
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(f(0)) → c22(F(ok(0)))
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0)))
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(g(0)) → c24(G(ok(0)))
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0)))
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0)))
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c26(S(ok(0)))
PROPER(sel(x0, f(z0))) → c27(SEL(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(sel(x0, cons(z0, z1))) → c27(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, g(z0))) → c27(SEL(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(sel(x0, s(z0))) → c27(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, sel(z0, z1))) → c27(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(f(z0), x1)) → c27(SEL(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c27(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(g(z0), x1)) → c27(SEL(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c27(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c27(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c27(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c27(SEL(ok(0), proper(x1)), PROPER(x1))
TOP(mark(f(z0))) → c28(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(cons(z0, z1))) → c28(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(g(z0))) → c28(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(sel(z0, z1))) → c28(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(0)) → c28(TOP(ok(0)))
K tuples:none
Defined Rule Symbols:

active, sel, f, cons, g, s, proper

Defined Pair Symbols:

ACTIVE, F, CONS, G, S, SEL, TOP, PROPER

Compound Symbols:

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

(45) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

TOP(mark(0)) → c28(TOP(ok(0)))
We considered the (Usable) Rules:

g(ok(z0)) → ok(g(z0))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(z0, z1)) → sel(active(z0), z1)
f(mark(z0)) → mark(f(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
f(ok(z0)) → ok(f(z0))
sel(mark(z0), z1) → mark(sel(z0, z1))
active(f(z0)) → f(active(z0))
g(mark(z0)) → mark(g(z0))
active(g(s(z0))) → mark(s(s(g(z0))))
active(g(0)) → mark(s(0))
active(f(z0)) → mark(cons(z0, f(g(z0))))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))
active(s(z0)) → s(active(z0))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
active(cons(z0, z1)) → cons(active(z0), z1)
sel(z0, mark(z1)) → mark(sel(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(g(z0)) → g(active(z0))
active(sel(z0, z1)) → sel(z0, active(z1))
And the Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0)))
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))))
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1))
ACTIVE(g(g(0))) → c7(G(mark(s(0))))
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(g(0))) → c8(S(mark(s(0))))
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0)))
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(f(0)) → c22(F(ok(0)))
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0)))
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(g(0)) → c24(G(ok(0)))
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0)))
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0)))
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c26(S(ok(0)))
PROPER(sel(x0, f(z0))) → c27(SEL(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(sel(x0, cons(z0, z1))) → c27(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, g(z0))) → c27(SEL(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(sel(x0, s(z0))) → c27(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, sel(z0, z1))) → c27(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(f(z0), x1)) → c27(SEL(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c27(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(g(z0), x1)) → c27(SEL(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c27(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c27(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c27(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c27(SEL(ok(0), proper(x1)), PROPER(x1))
TOP(mark(f(z0))) → c28(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(cons(z0, z1))) → c28(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(g(z0))) → c28(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(sel(z0, z1))) → c28(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(0)) → c28(TOP(ok(0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(ACTIVE(x1)) = 0   
POL(CONS(x1, x2)) = 0   
POL(F(x1)) = 0   
POL(G(x1)) = 0   
POL(PROPER(x1)) = 0   
POL(S(x1)) = 0   
POL(SEL(x1, x2)) = 0   
POL(TOP(x1)) = [2]x1   
POL(active(x1)) = x1   
POL(c(x1)) = x1   
POL(c10(x1, x2)) = x1 + x2   
POL(c11(x1)) = x1   
POL(c12(x1)) = x1   
POL(c13(x1)) = x1   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c16(x1)) = x1   
POL(c17(x1)) = x1   
POL(c18(x1)) = x1   
POL(c19(x1)) = x1   
POL(c2(x1)) = x1   
POL(c20(x1)) = x1   
POL(c21(x1)) = x1   
POL(c22(x1)) = x1   
POL(c22(x1, x2)) = x1 + x2   
POL(c23(x1, x2)) = x1 + x2   
POL(c23(x1, x2, x3)) = x1 + x2 + x3   
POL(c24(x1)) = x1   
POL(c24(x1, x2)) = x1 + x2   
POL(c26(x1)) = x1   
POL(c26(x1, x2)) = x1 + x2   
POL(c27(x1, x2)) = x1 + x2   
POL(c27(x1, x2, x3)) = x1 + x2 + x3   
POL(c28(x1)) = x1   
POL(c28(x1, x2)) = x1 + x2   
POL(c29(x1, x2)) = x1 + x2   
POL(c4(x1)) = x1   
POL(c5(x1)) = x1   
POL(c5(x1, x2)) = x1 + x2   
POL(c6(x1)) = x1   
POL(c6(x1, x2)) = x1 + x2   
POL(c7(x1)) = x1   
POL(c7(x1, x2)) = x1 + x2   
POL(c8(x1)) = x1   
POL(c8(x1, x2)) = x1 + x2   
POL(c9(x1, x2)) = x1 + x2   
POL(cons(x1, x2)) = [1]   
POL(f(x1)) = [1]   
POL(g(x1)) = [1]   
POL(mark(x1)) = [1]   
POL(ok(x1)) = x1   
POL(proper(x1)) = 0   
POL(s(x1)) = [1]   
POL(sel(x1, x2)) = [1]   

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0))))
active(g(0)) → mark(s(0))
active(g(s(z0))) → mark(s(s(g(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(f(z0)) → f(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(g(z0)) → g(active(z0))
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(f(z0)) → f(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(g(z0)) → g(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0)))
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))))
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1))
ACTIVE(g(g(0))) → c7(G(mark(s(0))))
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(g(0))) → c8(S(mark(s(0))))
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0)))
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(f(0)) → c22(F(ok(0)))
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0)))
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(g(0)) → c24(G(ok(0)))
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0)))
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0)))
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c26(S(ok(0)))
PROPER(sel(x0, f(z0))) → c27(SEL(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(sel(x0, cons(z0, z1))) → c27(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, g(z0))) → c27(SEL(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(sel(x0, s(z0))) → c27(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, sel(z0, z1))) → c27(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(f(z0), x1)) → c27(SEL(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c27(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(g(z0), x1)) → c27(SEL(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c27(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c27(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c27(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c27(SEL(ok(0), proper(x1)), PROPER(x1))
TOP(mark(f(z0))) → c28(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(cons(z0, z1))) → c28(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(g(z0))) → c28(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(sel(z0, z1))) → c28(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(0)) → c28(TOP(ok(0)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0)))
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))))
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1))
ACTIVE(g(g(0))) → c7(G(mark(s(0))))
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(g(0))) → c8(S(mark(s(0))))
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0)))
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(f(0)) → c22(F(ok(0)))
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0)))
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(g(0)) → c24(G(ok(0)))
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0)))
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0)))
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c26(S(ok(0)))
PROPER(sel(x0, f(z0))) → c27(SEL(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(sel(x0, cons(z0, z1))) → c27(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, g(z0))) → c27(SEL(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(sel(x0, s(z0))) → c27(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, sel(z0, z1))) → c27(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(f(z0), x1)) → c27(SEL(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c27(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(g(z0), x1)) → c27(SEL(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c27(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c27(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c27(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c27(SEL(ok(0), proper(x1)), PROPER(x1))
TOP(mark(f(z0))) → c28(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(cons(z0, z1))) → c28(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(g(z0))) → c28(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(sel(z0, z1))) → c28(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
K tuples:

TOP(mark(0)) → c28(TOP(ok(0)))
Defined Rule Symbols:

active, sel, f, cons, g, s, proper

Defined Pair Symbols:

ACTIVE, F, CONS, G, S, SEL, TOP, PROPER

Compound Symbols:

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

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

Use narrowing to replace TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(f(z0))) → c29(TOP(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
TOP(ok(g(0))) → c29(TOP(mark(s(0))), ACTIVE(g(0)))
TOP(ok(g(s(z0)))) → c29(TOP(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
TOP(ok(sel(0, cons(z0, z1)))) → c29(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
TOP(ok(sel(s(z0), cons(z1, z2)))) → c29(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
TOP(ok(f(z0))) → c29(TOP(f(active(z0))), ACTIVE(f(z0)))
TOP(ok(cons(z0, z1))) → c29(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(g(z0))) → c29(TOP(g(active(z0))), ACTIVE(g(z0)))
TOP(ok(s(z0))) → c29(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(sel(z0, z1))) → c29(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
TOP(ok(sel(z0, z1))) → c29(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0))))
active(g(0)) → mark(s(0))
active(g(s(z0))) → mark(s(s(g(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(f(z0)) → f(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(g(z0)) → g(active(z0))
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(f(z0)) → f(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(g(z0)) → g(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0)))
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))))
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1))
ACTIVE(g(g(0))) → c7(G(mark(s(0))))
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(g(0))) → c8(S(mark(s(0))))
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0)))
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(f(0)) → c22(F(ok(0)))
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0)))
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(g(0)) → c24(G(ok(0)))
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0)))
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0)))
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c26(S(ok(0)))
PROPER(sel(x0, f(z0))) → c27(SEL(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(sel(x0, cons(z0, z1))) → c27(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, g(z0))) → c27(SEL(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(sel(x0, s(z0))) → c27(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, sel(z0, z1))) → c27(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(f(z0), x1)) → c27(SEL(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c27(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(g(z0), x1)) → c27(SEL(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c27(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c27(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c27(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c27(SEL(ok(0), proper(x1)), PROPER(x1))
TOP(mark(f(z0))) → c28(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(cons(z0, z1))) → c28(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(g(z0))) → c28(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(sel(z0, z1))) → c28(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(mark(0)) → c28(TOP(ok(0)))
TOP(ok(f(z0))) → c29(TOP(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
TOP(ok(g(0))) → c29(TOP(mark(s(0))), ACTIVE(g(0)))
TOP(ok(g(s(z0)))) → c29(TOP(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
TOP(ok(sel(0, cons(z0, z1)))) → c29(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
TOP(ok(sel(s(z0), cons(z1, z2)))) → c29(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
TOP(ok(f(z0))) → c29(TOP(f(active(z0))), ACTIVE(f(z0)))
TOP(ok(cons(z0, z1))) → c29(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(g(z0))) → c29(TOP(g(active(z0))), ACTIVE(g(z0)))
TOP(ok(s(z0))) → c29(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(sel(z0, z1))) → c29(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
TOP(ok(sel(z0, z1))) → c29(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0)))
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))))
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1))
ACTIVE(g(g(0))) → c7(G(mark(s(0))))
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(g(0))) → c8(S(mark(s(0))))
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0)))
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(f(0)) → c22(F(ok(0)))
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0)))
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(g(0)) → c24(G(ok(0)))
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0)))
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0)))
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c26(S(ok(0)))
PROPER(sel(x0, f(z0))) → c27(SEL(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(sel(x0, cons(z0, z1))) → c27(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, g(z0))) → c27(SEL(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(sel(x0, s(z0))) → c27(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, sel(z0, z1))) → c27(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(f(z0), x1)) → c27(SEL(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c27(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(g(z0), x1)) → c27(SEL(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c27(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c27(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c27(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c27(SEL(ok(0), proper(x1)), PROPER(x1))
TOP(mark(f(z0))) → c28(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(cons(z0, z1))) → c28(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(g(z0))) → c28(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(sel(z0, z1))) → c28(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(ok(f(z0))) → c29(TOP(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
TOP(ok(g(0))) → c29(TOP(mark(s(0))), ACTIVE(g(0)))
TOP(ok(g(s(z0)))) → c29(TOP(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
TOP(ok(sel(0, cons(z0, z1)))) → c29(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
TOP(ok(sel(s(z0), cons(z1, z2)))) → c29(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
TOP(ok(f(z0))) → c29(TOP(f(active(z0))), ACTIVE(f(z0)))
TOP(ok(cons(z0, z1))) → c29(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(g(z0))) → c29(TOP(g(active(z0))), ACTIVE(g(z0)))
TOP(ok(s(z0))) → c29(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(sel(z0, z1))) → c29(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
TOP(ok(sel(z0, z1))) → c29(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
K tuples:

TOP(mark(0)) → c28(TOP(ok(0)))
Defined Rule Symbols:

active, sel, f, cons, g, s, proper

Defined Pair Symbols:

ACTIVE, F, CONS, G, S, SEL, PROPER, TOP

Compound Symbols:

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

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

Removed 1 trailing nodes:

TOP(mark(0)) → c28(TOP(ok(0)))

(50) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0))))
active(g(0)) → mark(s(0))
active(g(s(z0))) → mark(s(s(g(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(f(z0)) → f(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(g(z0)) → g(active(z0))
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(f(z0)) → f(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(g(z0)) → g(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0)))
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))))
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1))
ACTIVE(g(g(0))) → c7(G(mark(s(0))))
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(g(0))) → c8(S(mark(s(0))))
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0)))
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(f(0)) → c22(F(ok(0)))
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0)))
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(g(0)) → c24(G(ok(0)))
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0)))
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0)))
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c26(S(ok(0)))
PROPER(sel(x0, f(z0))) → c27(SEL(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(sel(x0, cons(z0, z1))) → c27(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, g(z0))) → c27(SEL(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(sel(x0, s(z0))) → c27(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, sel(z0, z1))) → c27(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(f(z0), x1)) → c27(SEL(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c27(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(g(z0), x1)) → c27(SEL(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c27(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c27(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c27(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c27(SEL(ok(0), proper(x1)), PROPER(x1))
TOP(mark(f(z0))) → c28(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(cons(z0, z1))) → c28(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(g(z0))) → c28(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(sel(z0, z1))) → c28(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(ok(f(z0))) → c29(TOP(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
TOP(ok(g(0))) → c29(TOP(mark(s(0))), ACTIVE(g(0)))
TOP(ok(g(s(z0)))) → c29(TOP(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
TOP(ok(sel(0, cons(z0, z1)))) → c29(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
TOP(ok(sel(s(z0), cons(z1, z2)))) → c29(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
TOP(ok(f(z0))) → c29(TOP(f(active(z0))), ACTIVE(f(z0)))
TOP(ok(cons(z0, z1))) → c29(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(g(z0))) → c29(TOP(g(active(z0))), ACTIVE(g(z0)))
TOP(ok(s(z0))) → c29(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(sel(z0, z1))) → c29(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
TOP(ok(sel(z0, z1))) → c29(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0)))
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))))
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1))
ACTIVE(g(g(0))) → c7(G(mark(s(0))))
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(g(0))) → c8(S(mark(s(0))))
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0)))
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(f(0)) → c22(F(ok(0)))
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0)))
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(g(0)) → c24(G(ok(0)))
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0)))
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0)))
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c26(S(ok(0)))
PROPER(sel(x0, f(z0))) → c27(SEL(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(sel(x0, cons(z0, z1))) → c27(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, g(z0))) → c27(SEL(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(sel(x0, s(z0))) → c27(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, sel(z0, z1))) → c27(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(f(z0), x1)) → c27(SEL(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c27(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(g(z0), x1)) → c27(SEL(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c27(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c27(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c27(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c27(SEL(ok(0), proper(x1)), PROPER(x1))
TOP(mark(f(z0))) → c28(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(cons(z0, z1))) → c28(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(g(z0))) → c28(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(sel(z0, z1))) → c28(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(ok(f(z0))) → c29(TOP(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
TOP(ok(g(0))) → c29(TOP(mark(s(0))), ACTIVE(g(0)))
TOP(ok(g(s(z0)))) → c29(TOP(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
TOP(ok(sel(0, cons(z0, z1)))) → c29(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
TOP(ok(sel(s(z0), cons(z1, z2)))) → c29(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
TOP(ok(f(z0))) → c29(TOP(f(active(z0))), ACTIVE(f(z0)))
TOP(ok(cons(z0, z1))) → c29(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(g(z0))) → c29(TOP(g(active(z0))), ACTIVE(g(z0)))
TOP(ok(s(z0))) → c29(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(sel(z0, z1))) → c29(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
TOP(ok(sel(z0, z1))) → c29(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, sel, f, cons, g, s, proper

Defined Pair Symbols:

ACTIVE, F, CONS, G, S, SEL, PROPER, TOP

Compound Symbols:

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

(51) CdtUnreachableProof (EQUIVALENT transformation)

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

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2))
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0))
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1))
ACTIVE(f(z0)) → c(G(z0))
ACTIVE(g(s(z0))) → c2(G(z0))
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0)))
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0))))
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1)))
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(f(g(0))) → c5(F(mark(s(0))))
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1))
ACTIVE(g(g(0))) → c7(G(mark(s(0))))
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))
ACTIVE(s(g(0))) → c8(S(mark(s(0))))
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0)))
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(f(0)) → c22(F(ok(0)))
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0)))
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(g(0)) → c24(G(ok(0)))
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0)))
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0)))
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
PROPER(s(0)) → c26(S(ok(0)))
PROPER(sel(x0, f(z0))) → c27(SEL(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0)))
PROPER(sel(x0, cons(z0, z1))) → c27(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(sel(x0, g(z0))) → c27(SEL(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(sel(x0, s(z0))) → c27(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(sel(x0, sel(z0, z1))) → c27(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1)))
PROPER(sel(f(z0), x1)) → c27(SEL(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1))
PROPER(sel(cons(z0, z1), x1)) → c27(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(sel(g(z0), x1)) → c27(SEL(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(sel(s(z0), x1)) → c27(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(sel(sel(z0, z1), x1)) → c27(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))
PROPER(sel(x0, 0)) → c27(SEL(proper(x0), ok(0)), PROPER(x0))
PROPER(sel(0, x1)) → c27(SEL(ok(0), proper(x1)), PROPER(x1))
TOP(mark(f(z0))) → c28(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(cons(z0, z1))) → c28(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(g(z0))) → c28(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(sel(z0, z1))) → c28(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))
TOP(ok(f(z0))) → c29(TOP(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0)))
TOP(ok(g(0))) → c29(TOP(mark(s(0))), ACTIVE(g(0)))
TOP(ok(g(s(z0)))) → c29(TOP(mark(s(s(g(z0))))), ACTIVE(g(s(z0))))
TOP(ok(sel(0, cons(z0, z1)))) → c29(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1))))
TOP(ok(sel(s(z0), cons(z1, z2)))) → c29(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2))))
TOP(ok(f(z0))) → c29(TOP(f(active(z0))), ACTIVE(f(z0)))
TOP(ok(cons(z0, z1))) → c29(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(g(z0))) → c29(TOP(g(active(z0))), ACTIVE(g(z0)))
TOP(ok(s(z0))) → c29(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(sel(z0, z1))) → c29(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1)))
TOP(ok(sel(z0, z1))) → c29(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))

(52) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0))))
active(g(0)) → mark(s(0))
active(g(s(z0))) → mark(s(s(g(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(f(z0)) → f(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(g(z0)) → g(active(z0))
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(f(z0)) → f(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(g(z0)) → g(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))
Tuples:

F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
S tuples:

F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
K tuples:none
Defined Rule Symbols:

active, sel, f, cons, g, s, proper

Defined Pair Symbols:

F, CONS, G, S, SEL

Compound Symbols:

c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21

(53) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

active(f(z0)) → mark(cons(z0, f(g(z0))))
active(g(0)) → mark(s(0))
active(g(s(z0))) → mark(s(s(g(z0))))
active(sel(0, cons(z0, z1))) → mark(z0)
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2))
active(f(z0)) → f(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(g(z0)) → g(active(z0))
active(s(z0)) → s(active(z0))
active(sel(z0, z1)) → sel(active(z0), z1)
active(sel(z0, z1)) → sel(z0, active(z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(f(z0)) → f(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(g(z0)) → g(proper(z0))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
proper(sel(z0, z1)) → sel(proper(z0), proper(z1))

(54) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
S tuples:

F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

F, CONS, G, S, SEL

Compound Symbols:

c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21

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

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

S(mark(z0)) → c17(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(CONS(x1, x2)) = [2]x2   
POL(F(x1)) = 0   
POL(G(x1)) = 0   
POL(S(x1)) = [2]x1   
POL(SEL(x1, x2)) = [3]x1   
POL(c11(x1)) = x1   
POL(c12(x1)) = x1   
POL(c13(x1)) = x1   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c16(x1)) = x1   
POL(c17(x1)) = x1   
POL(c18(x1)) = x1   
POL(c19(x1)) = x1   
POL(c20(x1)) = x1   
POL(c21(x1)) = x1   
POL(mark(x1)) = [2] + x1   
POL(ok(x1)) = x1   

(56) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
S tuples:

F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(ok(z0)) → c18(S(z0))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
K tuples:

S(mark(z0)) → c17(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
Defined Rule Symbols:none

Defined Pair Symbols:

F, CONS, G, S, SEL

Compound Symbols:

c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21

(57) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(ok(z0)) → c18(S(z0))
We considered the (Usable) Rules:none
And the Tuples:

F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(CONS(x1, x2)) = [3]x1 + [4]x2   
POL(F(x1)) = [4]x1   
POL(G(x1)) = [5]x1   
POL(S(x1)) = [5]x1   
POL(SEL(x1, x2)) = 0   
POL(c11(x1)) = x1   
POL(c12(x1)) = x1   
POL(c13(x1)) = x1   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c16(x1)) = x1   
POL(c17(x1)) = x1   
POL(c18(x1)) = x1   
POL(c19(x1)) = x1   
POL(c20(x1)) = x1   
POL(c21(x1)) = x1   
POL(mark(x1)) = [4] + x1   
POL(ok(x1)) = [5] + x1   

(58) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
S tuples:

SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
K tuples:

S(mark(z0)) → c17(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(ok(z0)) → c18(S(z0))
Defined Rule Symbols:none

Defined Pair Symbols:

F, CONS, G, S, SEL

Compound Symbols:

c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21

(59) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(CONS(x1, x2)) = [4]x2   
POL(F(x1)) = 0   
POL(G(x1)) = [4]x1   
POL(S(x1)) = 0   
POL(SEL(x1, x2)) = x1   
POL(c11(x1)) = x1   
POL(c12(x1)) = x1   
POL(c13(x1)) = x1   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c16(x1)) = x1   
POL(c17(x1)) = x1   
POL(c18(x1)) = x1   
POL(c19(x1)) = x1   
POL(c20(x1)) = x1   
POL(c21(x1)) = x1   
POL(mark(x1)) = [1] + x1   
POL(ok(x1)) = [1] + x1   

(60) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
S tuples:

SEL(z0, mark(z1)) → c20(SEL(z0, z1))
K tuples:

S(mark(z0)) → c17(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(ok(z0)) → c18(S(z0))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
Defined Rule Symbols:none

Defined Pair Symbols:

F, CONS, G, S, SEL

Compound Symbols:

c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21

(61) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)) transformation)

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

SEL(z0, mark(z1)) → c20(SEL(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(CONS(x1, x2)) = [2]x1 + [3]x12   
POL(F(x1)) = x12   
POL(G(x1)) = 0   
POL(S(x1)) = [2]x12   
POL(SEL(x1, x2)) = [3]x2 + x22   
POL(c11(x1)) = x1   
POL(c12(x1)) = x1   
POL(c13(x1)) = x1   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c16(x1)) = x1   
POL(c17(x1)) = x1   
POL(c18(x1)) = x1   
POL(c19(x1)) = x1   
POL(c20(x1)) = x1   
POL(c21(x1)) = x1   
POL(mark(x1)) = [1] + x1   
POL(ok(x1)) = x1   

(62) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(mark(z0)) → c17(S(z0))
S(ok(z0)) → c18(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
S tuples:none
K tuples:

S(mark(z0)) → c17(S(z0))
SEL(mark(z0), z1) → c19(SEL(z0, z1))
F(mark(z0)) → c11(F(z0))
F(ok(z0)) → c12(F(z0))
CONS(mark(z0), z1) → c13(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
G(mark(z0)) → c15(G(z0))
G(ok(z0)) → c16(G(z0))
S(ok(z0)) → c18(S(z0))
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
Defined Rule Symbols:none

Defined Pair Symbols:

F, CONS, G, S, SEL

Compound Symbols:

c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21

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

The set S is empty

(64) BOUNDS(1, 1)