211.86/72.11 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml

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The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^3)).

0 CpxTRS

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

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↳2 CdtProblem

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

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↳4 CdtProblem

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↳5 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))

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↳6 CdtProblem

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

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↳8 CdtProblem

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↳9 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))

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↳10 CdtProblem

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

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↳12 CdtProblem

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↳13 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))

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↳14 CdtProblem

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

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↳16 CdtProblem

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↳17 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))

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↳18 CdtProblem

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↳19 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))

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↳20 CdtProblem

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

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↳22 CdtProblem

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↳23 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))

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↳24 CdtProblem

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↳25 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))

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↳26 CdtProblem

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

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↳28 CdtProblem

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

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↳30 CdtProblem

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↳31 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))

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↳32 CdtProblem

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

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↳34 CdtProblem

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

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↳36 CdtProblem

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↳37 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))

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↳38 CdtProblem

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

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↳40 CdtProblem

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

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↳42 CdtProblem

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↳43 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))

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↳44 CdtProblem

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

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↳46 CdtProblem

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

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↳48 CdtProblem

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

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↳50 CdtProblem

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↳51 CdtNarrowingProof (BOTH BOUNDS(ID, ID))

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↳52 CdtProblem

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↳53 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))

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↳54 CdtProblem

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↳55 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))

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↳56 CdtProblem

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

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↳58 CdtProblem

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↳59 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))

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↳60 CdtProblem

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↳61 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))

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↳62 CdtProblem

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↳63 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))

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↳64 CdtProblem

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↳65 CdtNarrowingProof (BOTH BOUNDS(ID, ID))

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↳66 CdtProblem

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↳67 CdtUnreachableProof (⇔)

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↳68 CdtProblem

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↳69 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))

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↳70 CdtProblem

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↳71 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))

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↳72 CdtProblem

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↳73 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))

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↳74 CdtProblem

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↳75 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))

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↳76 CdtProblem

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↳77 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))

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↳78 CdtProblem

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↳79 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^3))))

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↳80 CdtProblem

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

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↳82 BOUNDS(O(1), O(1))

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(0) Obligation:

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

active(f(X)) → mark(cons(X, f(g(X)))) 212.23/72.24
active(g(0)) → mark(s(0)) 212.23/72.24
active(g(s(X))) → mark(s(s(g(X)))) 212.23/72.24
active(sel(0, cons(X, Y))) → mark(X) 212.23/72.24
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z)) 212.23/72.24
active(f(X)) → f(active(X)) 212.23/72.24
active(cons(X1, X2)) → cons(active(X1), X2) 212.23/72.24
active(g(X)) → g(active(X)) 212.23/72.24
active(s(X)) → s(active(X)) 212.23/72.24
active(sel(X1, X2)) → sel(active(X1), X2) 212.23/72.24
active(sel(X1, X2)) → sel(X1, active(X2)) 212.23/72.24
f(mark(X)) → mark(f(X)) 212.23/72.24
cons(mark(X1), X2) → mark(cons(X1, X2)) 212.23/72.24
g(mark(X)) → mark(g(X)) 212.23/72.24
s(mark(X)) → mark(s(X)) 212.23/72.24
sel(mark(X1), X2) → mark(sel(X1, X2)) 212.23/72.24
sel(X1, mark(X2)) → mark(sel(X1, X2)) 212.23/72.24
proper(f(X)) → f(proper(X)) 212.23/72.24
proper(cons(X1, X2)) → cons(proper(X1), proper(X2)) 212.23/72.24
proper(g(X)) → g(proper(X)) 212.23/72.24
proper(0) → ok(0) 212.23/72.24
proper(s(X)) → s(proper(X)) 212.23/72.24
proper(sel(X1, X2)) → sel(proper(X1), proper(X2)) 212.23/72.24
f(ok(X)) → ok(f(X)) 212.23/72.24
cons(ok(X1), ok(X2)) → ok(cons(X1, X2)) 212.23/72.24
g(ok(X)) → ok(g(X)) 212.23/72.24
s(ok(X)) → ok(s(X)) 212.23/72.24
sel(ok(X1), ok(X2)) → ok(sel(X1, X2)) 212.23/72.24
top(mark(X)) → top(proper(X)) 212.23/72.24
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

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(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0)))) 212.23/72.24
active(g(0)) → mark(s(0)) 212.23/72.24
active(g(s(z0))) → mark(s(s(g(z0)))) 212.23/72.24
active(sel(0, cons(z0, z1))) → mark(z0) 212.23/72.24
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 212.23/72.24
active(f(z0)) → f(active(z0)) 212.23/72.24
active(cons(z0, z1)) → cons(active(z0), z1) 212.23/72.24
active(g(z0)) → g(active(z0)) 212.23/72.24
active(s(z0)) → s(active(z0)) 212.23/72.24
active(sel(z0, z1)) → sel(active(z0), z1) 212.23/72.24
active(sel(z0, z1)) → sel(z0, active(z1)) 212.23/72.24
f(mark(z0)) → mark(f(z0)) 212.23/72.24
f(ok(z0)) → ok(f(z0)) 212.23/72.24
cons(mark(z0), z1) → mark(cons(z0, z1)) 212.23/72.24
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 212.23/72.24
g(mark(z0)) → mark(g(z0)) 212.23/72.24
g(ok(z0)) → ok(g(z0)) 212.23/72.24
s(mark(z0)) → mark(s(z0)) 212.23/72.24
s(ok(z0)) → ok(s(z0)) 212.23/72.24
sel(mark(z0), z1) → mark(sel(z0, z1)) 212.23/72.24
sel(z0, mark(z1)) → mark(sel(z0, z1)) 212.23/72.24
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 212.23/72.24
proper(f(z0)) → f(proper(z0)) 212.23/72.24
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 212.23/72.24
proper(g(z0)) → g(proper(z0)) 212.23/72.24
proper(0) → ok(0) 212.23/72.24
proper(s(z0)) → s(proper(z0)) 212.23/72.24
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 212.23/72.24
top(mark(z0)) → top(proper(z0)) 212.23/72.24
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(f(z0)) → c(CONS(z0, f(g(z0))), F(g(z0)), G(z0)) 212.23/72.24
ACTIVE(g(0)) → c1(S(0)) 212.23/72.24
ACTIVE(g(s(z0))) → c2(S(s(g(z0))), S(g(z0)), G(z0)) 212.23/72.24
ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.23/72.24
ACTIVE(f(z0)) → c5(F(active(z0)), ACTIVE(z0)) 212.23/72.24
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0)) 212.23/72.24
ACTIVE(g(z0)) → c7(G(active(z0)), ACTIVE(z0)) 212.23/72.24
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0)) 212.23/72.24
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.23/72.24
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.23/72.24
F(mark(z0)) → c11(F(z0)) 212.23/72.24
F(ok(z0)) → c12(F(z0)) 212.23/72.24
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.23/72.24
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.23/72.24
G(mark(z0)) → c15(G(z0)) 212.23/72.24
G(ok(z0)) → c16(G(z0)) 212.23/72.24
S(mark(z0)) → c17(S(z0)) 212.23/72.24
S(ok(z0)) → c18(S(z0)) 212.23/72.24
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.23/72.24
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.23/72.24
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.23/72.24
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0)) 212.23/72.24
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.23/72.24
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0)) 212.23/72.24
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.23/72.24
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.23/72.24
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.23/72.24
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)) 212.23/72.24
ACTIVE(g(0)) → c1(S(0)) 212.23/72.24
ACTIVE(g(s(z0))) → c2(S(s(g(z0))), S(g(z0)), G(z0)) 212.23/72.24
ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.23/72.24
ACTIVE(f(z0)) → c5(F(active(z0)), ACTIVE(z0)) 212.23/72.24
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0)) 212.23/72.24
ACTIVE(g(z0)) → c7(G(active(z0)), ACTIVE(z0)) 212.23/72.24
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0)) 212.23/72.24
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.23/72.24
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.23/72.24
F(mark(z0)) → c11(F(z0)) 212.23/72.24
F(ok(z0)) → c12(F(z0)) 212.23/72.24
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.23/72.24
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.23/72.24
G(mark(z0)) → c15(G(z0)) 212.23/72.24
G(ok(z0)) → c16(G(z0)) 212.23/72.24
S(mark(z0)) → c17(S(z0)) 212.23/72.24
S(ok(z0)) → c18(S(z0)) 212.23/72.24
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.23/72.24
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.23/72.24
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.23/72.24
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0)) 212.23/72.24
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.23/72.24
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0)) 212.23/72.24
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.23/72.24
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.23/72.24
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.23/72.24
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, 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

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

Removed 5 trailing tuple parts

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(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0)))) 212.23/72.24
active(g(0)) → mark(s(0)) 212.23/72.24
active(g(s(z0))) → mark(s(s(g(z0)))) 212.23/72.24
active(sel(0, cons(z0, z1))) → mark(z0) 212.23/72.24
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 212.23/72.24
active(f(z0)) → f(active(z0)) 212.23/72.24
active(cons(z0, z1)) → cons(active(z0), z1) 212.23/72.24
active(g(z0)) → g(active(z0)) 212.23/72.24
active(s(z0)) → s(active(z0)) 212.23/72.24
active(sel(z0, z1)) → sel(active(z0), z1) 212.23/72.24
active(sel(z0, z1)) → sel(z0, active(z1)) 212.23/72.24
f(mark(z0)) → mark(f(z0)) 212.23/72.24
f(ok(z0)) → ok(f(z0)) 212.23/72.24
cons(mark(z0), z1) → mark(cons(z0, z1)) 212.23/72.24
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 212.23/72.24
g(mark(z0)) → mark(g(z0)) 212.23/72.24
g(ok(z0)) → ok(g(z0)) 212.23/72.24
s(mark(z0)) → mark(s(z0)) 212.23/72.24
s(ok(z0)) → ok(s(z0)) 212.23/72.24
sel(mark(z0), z1) → mark(sel(z0, z1)) 212.23/72.24
sel(z0, mark(z1)) → mark(sel(z0, z1)) 212.23/72.24
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 212.23/72.24
proper(f(z0)) → f(proper(z0)) 212.23/72.24
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 212.23/72.24
proper(g(z0)) → g(proper(z0)) 212.23/72.24
proper(0) → ok(0) 212.23/72.24
proper(s(z0)) → s(proper(z0)) 212.23/72.24
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 212.23/72.24
top(mark(z0)) → top(proper(z0)) 212.23/72.24
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.23/72.24
ACTIVE(f(z0)) → c5(F(active(z0)), ACTIVE(z0)) 212.23/72.24
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0)) 212.23/72.24
ACTIVE(g(z0)) → c7(G(active(z0)), ACTIVE(z0)) 212.23/72.24
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0)) 212.23/72.24
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.23/72.24
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.23/72.24
F(mark(z0)) → c11(F(z0)) 212.23/72.24
F(ok(z0)) → c12(F(z0)) 212.23/72.24
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.23/72.24
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.23/72.24
G(mark(z0)) → c15(G(z0)) 212.23/72.24
G(ok(z0)) → c16(G(z0)) 212.23/72.24
S(mark(z0)) → c17(S(z0)) 212.23/72.24
S(ok(z0)) → c18(S(z0)) 212.23/72.24
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.23/72.24
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.23/72.24
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.23/72.24
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0)) 212.23/72.24
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.23/72.24
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0)) 212.23/72.24
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.23/72.24
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.23/72.24
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.23/72.24
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.23/72.24
ACTIVE(f(z0)) → c(G(z0)) 212.23/72.24
ACTIVE(g(0)) → c1 212.23/72.24
ACTIVE(g(s(z0))) → c2(G(z0))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.23/72.24
ACTIVE(f(z0)) → c5(F(active(z0)), ACTIVE(z0)) 212.23/72.24
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0)) 212.23/72.24
ACTIVE(g(z0)) → c7(G(active(z0)), ACTIVE(z0)) 212.23/72.24
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0)) 212.23/72.24
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.23/72.24
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.23/72.24
F(mark(z0)) → c11(F(z0)) 212.23/72.24
F(ok(z0)) → c12(F(z0)) 212.23/72.24
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.23/72.24
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.23/72.24
G(mark(z0)) → c15(G(z0)) 212.23/72.24
G(ok(z0)) → c16(G(z0)) 212.23/72.24
S(mark(z0)) → c17(S(z0)) 212.23/72.24
S(ok(z0)) → c18(S(z0)) 212.23/72.24
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.23/72.24
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.23/72.24
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.23/72.24
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0)) 212.23/72.24
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.23/72.24
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0)) 212.23/72.24
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.23/72.24
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.23/72.24
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.23/72.24
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.23/72.24
ACTIVE(f(z0)) → c(G(z0)) 212.23/72.24
ACTIVE(g(0)) → c1 212.23/72.24
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, c1, c2

212.23/72.24

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

Removed 1 trailing nodes:

ACTIVE(g(0)) → c1

212.23/72.24

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0)))) 212.23/72.24
active(g(0)) → mark(s(0)) 212.23/72.24
active(g(s(z0))) → mark(s(s(g(z0)))) 212.23/72.24
active(sel(0, cons(z0, z1))) → mark(z0) 212.23/72.24
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 212.23/72.24
active(f(z0)) → f(active(z0)) 212.23/72.24
active(cons(z0, z1)) → cons(active(z0), z1) 212.23/72.24
active(g(z0)) → g(active(z0)) 212.23/72.24
active(s(z0)) → s(active(z0)) 212.23/72.24
active(sel(z0, z1)) → sel(active(z0), z1) 212.23/72.25
active(sel(z0, z1)) → sel(z0, active(z1)) 212.23/72.25
f(mark(z0)) → mark(f(z0)) 212.23/72.25
f(ok(z0)) → ok(f(z0)) 212.23/72.25
cons(mark(z0), z1) → mark(cons(z0, z1)) 212.23/72.26
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 212.23/72.26
g(mark(z0)) → mark(g(z0)) 212.23/72.26
g(ok(z0)) → ok(g(z0)) 212.23/72.26
s(mark(z0)) → mark(s(z0)) 212.23/72.26
s(ok(z0)) → ok(s(z0)) 212.23/72.26
sel(mark(z0), z1) → mark(sel(z0, z1)) 212.23/72.26
sel(z0, mark(z1)) → mark(sel(z0, z1)) 212.23/72.26
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 212.23/72.26
proper(f(z0)) → f(proper(z0)) 212.23/72.26
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 212.23/72.26
proper(g(z0)) → g(proper(z0)) 212.23/72.26
proper(0) → ok(0) 212.23/72.26
proper(s(z0)) → s(proper(z0)) 212.23/72.26
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 212.23/72.26
top(mark(z0)) → top(proper(z0)) 212.23/72.26
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.23/72.26
ACTIVE(f(z0)) → c5(F(active(z0)), ACTIVE(z0)) 212.23/72.26
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0)) 212.23/72.26
ACTIVE(g(z0)) → c7(G(active(z0)), ACTIVE(z0)) 212.23/72.26
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0)) 212.23/72.26
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.23/72.26
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.23/72.26
F(mark(z0)) → c11(F(z0)) 212.23/72.26
F(ok(z0)) → c12(F(z0)) 212.23/72.26
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.23/72.26
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.23/72.26
G(mark(z0)) → c15(G(z0)) 212.23/72.26
G(ok(z0)) → c16(G(z0)) 212.23/72.26
S(mark(z0)) → c17(S(z0)) 212.23/72.26
S(ok(z0)) → c18(S(z0)) 212.23/72.26
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.23/72.26
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.23/72.26
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.23/72.26
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0)) 212.23/72.26
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.23/72.26
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0)) 212.23/72.26
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.23/72.26
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.23/72.26
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.23/72.26
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.23/72.26
ACTIVE(f(z0)) → c(G(z0)) 212.23/72.26
ACTIVE(g(0)) → c1 212.23/72.26
ACTIVE(g(s(z0))) → c2(G(z0))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.23/72.26
ACTIVE(f(z0)) → c5(F(active(z0)), ACTIVE(z0)) 212.23/72.26
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0)) 212.23/72.26
ACTIVE(g(z0)) → c7(G(active(z0)), ACTIVE(z0)) 212.23/72.26
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0)) 212.23/72.26
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.23/72.26
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.23/72.26
F(mark(z0)) → c11(F(z0)) 212.23/72.26
F(ok(z0)) → c12(F(z0)) 212.23/72.26
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.23/72.26
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.23/72.26
G(mark(z0)) → c15(G(z0)) 212.23/72.26
G(ok(z0)) → c16(G(z0)) 212.23/72.26
S(mark(z0)) → c17(S(z0)) 212.23/72.26
S(ok(z0)) → c18(S(z0)) 212.23/72.26
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.23/72.26
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.23/72.26
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.23/72.26
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0)) 212.23/72.26
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.23/72.26
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0)) 212.23/72.26
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.23/72.26
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.23/72.26
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.23/72.26
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.23/72.26
ACTIVE(f(z0)) → c(G(z0)) 212.23/72.26
ACTIVE(g(0)) → c1 212.23/72.26
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, c1, c2

212.23/72.26

(7) 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))) 212.23/72.26
ACTIVE(f(g(0))) → c5(F(mark(s(0))), ACTIVE(g(0))) 212.23/72.26
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.23/72.26
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.23/72.26
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.23/72.26
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.23/72.26
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.23/72.26
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.23/72.26
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.23/72.26
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.23/72.26
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))

212.23/72.26

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0)))) 212.23/72.26
active(g(0)) → mark(s(0)) 212.23/72.26
active(g(s(z0))) → mark(s(s(g(z0)))) 212.23/72.26
active(sel(0, cons(z0, z1))) → mark(z0) 212.23/72.26
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 212.23/72.26
active(f(z0)) → f(active(z0)) 212.23/72.26
active(cons(z0, z1)) → cons(active(z0), z1) 212.23/72.26
active(g(z0)) → g(active(z0)) 212.23/72.26
active(s(z0)) → s(active(z0)) 212.23/72.26
active(sel(z0, z1)) → sel(active(z0), z1) 212.23/72.26
active(sel(z0, z1)) → sel(z0, active(z1)) 212.23/72.26
f(mark(z0)) → mark(f(z0)) 212.23/72.26
f(ok(z0)) → ok(f(z0)) 212.23/72.26
cons(mark(z0), z1) → mark(cons(z0, z1)) 212.23/72.26
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 212.23/72.26
g(mark(z0)) → mark(g(z0)) 212.23/72.26
g(ok(z0)) → ok(g(z0)) 212.23/72.26
s(mark(z0)) → mark(s(z0)) 212.23/72.26
s(ok(z0)) → ok(s(z0)) 212.23/72.26
sel(mark(z0), z1) → mark(sel(z0, z1)) 212.23/72.26
sel(z0, mark(z1)) → mark(sel(z0, z1)) 212.23/72.26
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 212.23/72.26
proper(f(z0)) → f(proper(z0)) 212.23/72.26
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 212.23/72.26
proper(g(z0)) → g(proper(z0)) 212.23/72.26
proper(0) → ok(0) 212.23/72.26
proper(s(z0)) → s(proper(z0)) 212.23/72.26
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 212.23/72.26
top(mark(z0)) → top(proper(z0)) 212.23/72.26
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.23/72.26
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0)) 212.23/72.26
ACTIVE(g(z0)) → c7(G(active(z0)), ACTIVE(z0)) 212.23/72.26
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0)) 212.23/72.26
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.23/72.26
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.23/72.26
F(mark(z0)) → c11(F(z0)) 212.23/72.26
F(ok(z0)) → c12(F(z0)) 212.23/72.26
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.23/72.26
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.23/72.26
G(mark(z0)) → c15(G(z0)) 212.23/72.26
G(ok(z0)) → c16(G(z0)) 212.23/72.26
S(mark(z0)) → c17(S(z0)) 212.23/72.26
S(ok(z0)) → c18(S(z0)) 212.23/72.26
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.23/72.26
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.23/72.26
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.23/72.26
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0)) 212.23/72.26
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.23/72.26
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0)) 212.23/72.26
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.23/72.26
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.23/72.26
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.23/72.26
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.23/72.26
ACTIVE(f(z0)) → c(G(z0)) 212.23/72.26
ACTIVE(g(0)) → c1 212.23/72.26
ACTIVE(g(s(z0))) → c2(G(z0)) 212.23/72.26
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.23/72.26
ACTIVE(f(g(0))) → c5(F(mark(s(0))), ACTIVE(g(0))) 212.23/72.26
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.23/72.26
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.23/72.26
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.23/72.26
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.23/72.26
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.23/72.26
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.23/72.26
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.23/72.26
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.23/72.26
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)) 212.23/72.26
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0)) 212.23/72.26
ACTIVE(g(z0)) → c7(G(active(z0)), ACTIVE(z0)) 212.23/72.26
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0)) 212.23/72.26
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.23/72.26
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.23/72.26
F(mark(z0)) → c11(F(z0)) 212.23/72.26
F(ok(z0)) → c12(F(z0)) 212.23/72.26
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.23/72.26
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.23/72.26
G(mark(z0)) → c15(G(z0)) 212.23/72.26
G(ok(z0)) → c16(G(z0)) 212.23/72.26
S(mark(z0)) → c17(S(z0)) 212.23/72.26
S(ok(z0)) → c18(S(z0)) 212.23/72.26
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.23/72.26
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.23/72.26
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.23/72.26
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0)) 212.23/72.26
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.23/72.26
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0)) 212.23/72.26
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.23/72.26
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.23/72.26
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.23/72.26
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.23/72.26
ACTIVE(f(z0)) → c(G(z0)) 212.23/72.26
ACTIVE(g(0)) → c1 212.23/72.26
ACTIVE(g(s(z0))) → c2(G(z0)) 212.23/72.26
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.23/72.26
ACTIVE(f(g(0))) → c5(F(mark(s(0))), ACTIVE(g(0))) 212.23/72.26
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.23/72.26
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.23/72.26
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.23/72.26
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.23/72.26
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.23/72.26
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.23/72.26
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.23/72.26
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.23/72.26
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))

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

212.23/72.26

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

Removed 1 trailing nodes:

ACTIVE(g(0)) → c1

212.23/72.26

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0)))) 212.23/72.26
active(g(0)) → mark(s(0)) 212.23/72.26
active(g(s(z0))) → mark(s(s(g(z0)))) 212.23/72.26
active(sel(0, cons(z0, z1))) → mark(z0) 212.23/72.26
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 212.23/72.26
active(f(z0)) → f(active(z0)) 212.23/72.26
active(cons(z0, z1)) → cons(active(z0), z1) 212.23/72.26
active(g(z0)) → g(active(z0)) 212.23/72.26
active(s(z0)) → s(active(z0)) 212.23/72.26
active(sel(z0, z1)) → sel(active(z0), z1) 212.23/72.26
active(sel(z0, z1)) → sel(z0, active(z1)) 212.23/72.26
f(mark(z0)) → mark(f(z0)) 212.23/72.26
f(ok(z0)) → ok(f(z0)) 212.23/72.26
cons(mark(z0), z1) → mark(cons(z0, z1)) 212.23/72.26
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 212.23/72.26
g(mark(z0)) → mark(g(z0)) 212.23/72.26
g(ok(z0)) → ok(g(z0)) 212.23/72.26
s(mark(z0)) → mark(s(z0)) 212.23/72.26
s(ok(z0)) → ok(s(z0)) 212.23/72.26
sel(mark(z0), z1) → mark(sel(z0, z1)) 212.23/72.26
sel(z0, mark(z1)) → mark(sel(z0, z1)) 212.23/72.26
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 212.23/72.26
proper(f(z0)) → f(proper(z0)) 212.23/72.26
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 212.23/72.26
proper(g(z0)) → g(proper(z0)) 212.23/72.26
proper(0) → ok(0) 212.23/72.26
proper(s(z0)) → s(proper(z0)) 212.23/72.26
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 212.23/72.26
top(mark(z0)) → top(proper(z0)) 212.23/72.26
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.23/72.26
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0)) 212.23/72.26
ACTIVE(g(z0)) → c7(G(active(z0)), ACTIVE(z0)) 212.23/72.26
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0)) 212.23/72.26
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.23/72.26
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.23/72.26
F(mark(z0)) → c11(F(z0)) 212.23/72.26
F(ok(z0)) → c12(F(z0)) 212.23/72.26
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.23/72.26
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.23/72.26
G(mark(z0)) → c15(G(z0)) 212.23/72.26
G(ok(z0)) → c16(G(z0)) 212.23/72.26
S(mark(z0)) → c17(S(z0)) 212.23/72.26
S(ok(z0)) → c18(S(z0)) 212.23/72.26
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.23/72.26
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.23/72.26
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.23/72.26
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0)) 212.23/72.26
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.23/72.26
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0)) 212.23/72.26
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.23/72.26
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.23/72.26
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.23/72.26
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.23/72.26
ACTIVE(f(z0)) → c(G(z0)) 212.23/72.26
ACTIVE(g(0)) → c1 212.23/72.26
ACTIVE(g(s(z0))) → c2(G(z0)) 212.23/72.26
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.23/72.26
ACTIVE(f(g(0))) → c5(F(mark(s(0))), ACTIVE(g(0))) 212.23/72.26
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.23/72.26
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.23/72.26
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.23/72.26
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.23/72.26
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.23/72.26
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.23/72.26
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.23/72.26
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.23/72.26
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)) 212.23/72.26
ACTIVE(cons(z0, z1)) → c6(CONS(active(z0), z1), ACTIVE(z0)) 212.23/72.26
ACTIVE(g(z0)) → c7(G(active(z0)), ACTIVE(z0)) 212.23/72.26
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0)) 212.23/72.26
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.23/72.26
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.23/72.26
F(mark(z0)) → c11(F(z0)) 212.23/72.26
F(ok(z0)) → c12(F(z0)) 212.23/72.26
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.23/72.26
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.23/72.26
G(mark(z0)) → c15(G(z0)) 212.23/72.26
G(ok(z0)) → c16(G(z0)) 212.23/72.26
S(mark(z0)) → c17(S(z0)) 212.23/72.26
S(ok(z0)) → c18(S(z0)) 212.23/72.26
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.23/72.26
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.23/72.26
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.23/72.26
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0)) 212.23/72.26
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.23/72.26
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0)) 212.23/72.26
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.23/72.26
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.23/72.26
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.23/72.26
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.23/72.26
ACTIVE(f(z0)) → c(G(z0)) 212.23/72.26
ACTIVE(g(0)) → c1 212.23/72.26
ACTIVE(g(s(z0))) → c2(G(z0)) 212.23/72.26
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.23/72.26
ACTIVE(f(g(0))) → c5(F(mark(s(0))), ACTIVE(g(0))) 212.23/72.26
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.23/72.26
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.23/72.26
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.23/72.26
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.23/72.26
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.23/72.26
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.23/72.26
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.23/72.26
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.23/72.26
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))

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

212.23/72.26

(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))) 212.23/72.26
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1), ACTIVE(g(0))) 212.23/72.26
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.23/72.26
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.23/72.26
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.23/72.26
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.23/72.26
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.23/72.26
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.23/72.26
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.23/72.26
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.23/72.26
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))

212.23/72.26

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0)))) 212.23/72.26
active(g(0)) → mark(s(0)) 212.23/72.26
active(g(s(z0))) → mark(s(s(g(z0)))) 212.23/72.26
active(sel(0, cons(z0, z1))) → mark(z0) 212.23/72.26
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 212.23/72.26
active(f(z0)) → f(active(z0)) 212.23/72.26
active(cons(z0, z1)) → cons(active(z0), z1) 212.23/72.26
active(g(z0)) → g(active(z0)) 212.23/72.26
active(s(z0)) → s(active(z0)) 212.23/72.26
active(sel(z0, z1)) → sel(active(z0), z1) 212.23/72.26
active(sel(z0, z1)) → sel(z0, active(z1)) 212.23/72.26
f(mark(z0)) → mark(f(z0)) 212.23/72.26
f(ok(z0)) → ok(f(z0)) 212.23/72.26
cons(mark(z0), z1) → mark(cons(z0, z1)) 212.23/72.26
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 212.23/72.26
g(mark(z0)) → mark(g(z0)) 212.23/72.26
g(ok(z0)) → ok(g(z0)) 212.23/72.26
s(mark(z0)) → mark(s(z0)) 212.23/72.26
s(ok(z0)) → ok(s(z0)) 212.23/72.26
sel(mark(z0), z1) → mark(sel(z0, z1)) 212.23/72.26
sel(z0, mark(z1)) → mark(sel(z0, z1)) 212.23/72.26
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 212.23/72.26
proper(f(z0)) → f(proper(z0)) 212.23/72.26
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 212.23/72.26
proper(g(z0)) → g(proper(z0)) 212.23/72.26
proper(0) → ok(0) 212.23/72.26
proper(s(z0)) → s(proper(z0)) 212.23/72.26
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 212.23/72.26
top(mark(z0)) → top(proper(z0)) 212.23/72.26
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.23/72.26
ACTIVE(g(z0)) → c7(G(active(z0)), ACTIVE(z0)) 212.23/72.26
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0)) 212.23/72.26
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.23/72.26
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.23/72.26
F(mark(z0)) → c11(F(z0)) 212.23/72.26
F(ok(z0)) → c12(F(z0)) 212.23/72.26
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.23/72.26
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.23/72.26
G(mark(z0)) → c15(G(z0)) 212.23/72.26
G(ok(z0)) → c16(G(z0)) 212.23/72.26
S(mark(z0)) → c17(S(z0)) 212.23/72.26
S(ok(z0)) → c18(S(z0)) 212.23/72.26
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.23/72.26
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.23/72.26
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.23/72.26
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0)) 212.23/72.26
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.23/72.26
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0)) 212.23/72.26
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.23/72.26
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.23/72.26
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.23/72.26
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.23/72.26
ACTIVE(f(z0)) → c(G(z0)) 212.23/72.26
ACTIVE(g(0)) → c1 212.23/72.26
ACTIVE(g(s(z0))) → c2(G(z0)) 212.23/72.26
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.23/72.26
ACTIVE(f(g(0))) → c5(F(mark(s(0))), ACTIVE(g(0))) 212.23/72.26
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.23/72.26
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.23/72.26
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.23/72.26
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.23/72.26
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.23/72.26
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.23/72.26
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.23/72.26
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.23/72.26
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.23/72.26
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.23/72.26
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1), ACTIVE(g(0))) 212.23/72.26
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.23/72.26
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.23/72.26
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.23/72.26
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.23/72.26
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.23/72.26
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.23/72.26
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.23/72.26
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.23/72.26
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)) 212.23/72.26
ACTIVE(g(z0)) → c7(G(active(z0)), ACTIVE(z0)) 212.23/72.26
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0)) 212.23/72.26
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.23/72.26
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.23/72.26
F(mark(z0)) → c11(F(z0)) 212.23/72.26
F(ok(z0)) → c12(F(z0)) 212.23/72.26
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.23/72.26
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.23/72.26
G(mark(z0)) → c15(G(z0)) 212.23/72.26
G(ok(z0)) → c16(G(z0)) 212.23/72.26
S(mark(z0)) → c17(S(z0)) 212.23/72.26
S(ok(z0)) → c18(S(z0)) 212.23/72.26
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.23/72.26
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.23/72.26
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.23/72.26
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0)) 212.23/72.26
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.23/72.26
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0)) 212.23/72.26
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.23/72.26
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.23/72.26
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.23/72.26
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.23/72.26
ACTIVE(f(z0)) → c(G(z0)) 212.23/72.26
ACTIVE(g(0)) → c1 212.23/72.26
ACTIVE(g(s(z0))) → c2(G(z0)) 212.23/72.26
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.23/72.26
ACTIVE(f(g(0))) → c5(F(mark(s(0))), ACTIVE(g(0))) 212.23/72.26
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.23/72.26
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.23/72.26
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.23/72.26
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.23/72.26
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.23/72.26
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.23/72.26
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.23/72.26
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.23/72.26
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.23/72.26
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.23/72.26
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1), ACTIVE(g(0))) 212.23/72.26
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.23/72.26
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.23/72.26
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.23/72.26
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.23/72.26
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.23/72.26
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.23/72.26
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.23/72.26
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.23/72.26
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))

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

212.23/72.26

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

Removed 1 trailing nodes:

ACTIVE(g(0)) → c1

212.23/72.26

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0)))) 212.23/72.26
active(g(0)) → mark(s(0)) 212.23/72.26
active(g(s(z0))) → mark(s(s(g(z0)))) 212.23/72.26
active(sel(0, cons(z0, z1))) → mark(z0) 212.23/72.26
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 212.23/72.26
active(f(z0)) → f(active(z0)) 212.23/72.26
active(cons(z0, z1)) → cons(active(z0), z1) 212.23/72.26
active(g(z0)) → g(active(z0)) 212.23/72.26
active(s(z0)) → s(active(z0)) 212.23/72.26
active(sel(z0, z1)) → sel(active(z0), z1) 212.23/72.26
active(sel(z0, z1)) → sel(z0, active(z1)) 212.23/72.26
f(mark(z0)) → mark(f(z0)) 212.23/72.26
f(ok(z0)) → ok(f(z0)) 212.23/72.26
cons(mark(z0), z1) → mark(cons(z0, z1)) 212.23/72.26
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 212.23/72.26
g(mark(z0)) → mark(g(z0)) 212.23/72.26
g(ok(z0)) → ok(g(z0)) 212.23/72.26
s(mark(z0)) → mark(s(z0)) 212.23/72.26
s(ok(z0)) → ok(s(z0)) 212.23/72.26
sel(mark(z0), z1) → mark(sel(z0, z1)) 212.23/72.26
sel(z0, mark(z1)) → mark(sel(z0, z1)) 212.23/72.26
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 212.23/72.26
proper(f(z0)) → f(proper(z0)) 212.23/72.26
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 212.23/72.26
proper(g(z0)) → g(proper(z0)) 212.23/72.26
proper(0) → ok(0) 212.23/72.26
proper(s(z0)) → s(proper(z0)) 212.23/72.26
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 212.23/72.26
top(mark(z0)) → top(proper(z0)) 212.23/72.26
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.23/72.26
ACTIVE(g(z0)) → c7(G(active(z0)), ACTIVE(z0)) 212.23/72.26
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0)) 212.23/72.26
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.23/72.26
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.23/72.26
F(mark(z0)) → c11(F(z0)) 212.23/72.26
F(ok(z0)) → c12(F(z0)) 212.23/72.26
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.23/72.26
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.23/72.26
G(mark(z0)) → c15(G(z0)) 212.23/72.26
G(ok(z0)) → c16(G(z0)) 212.23/72.26
S(mark(z0)) → c17(S(z0)) 212.23/72.26
S(ok(z0)) → c18(S(z0)) 212.23/72.26
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.23/72.26
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.23/72.26
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.23/72.26
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0)) 212.23/72.26
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.23/72.26
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0)) 212.23/72.26
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.23/72.26
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.23/72.26
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.23/72.26
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.23/72.26
ACTIVE(f(z0)) → c(G(z0)) 212.23/72.26
ACTIVE(g(0)) → c1 212.23/72.26
ACTIVE(g(s(z0))) → c2(G(z0)) 212.23/72.26
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.23/72.26
ACTIVE(f(g(0))) → c5(F(mark(s(0))), ACTIVE(g(0))) 212.23/72.26
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.23/72.26
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.23/72.26
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.23/72.26
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.23/72.26
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.23/72.26
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.23/72.26
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.23/72.26
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.23/72.26
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.23/72.26
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.23/72.26
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1), ACTIVE(g(0))) 212.23/72.26
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.23/72.26
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.23/72.26
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.23/72.26
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.23/72.26
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.23/72.26
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.23/72.26
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.23/72.26
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.23/72.26
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)) 212.23/72.26
ACTIVE(g(z0)) → c7(G(active(z0)), ACTIVE(z0)) 212.23/72.26
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0)) 212.23/72.26
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.23/72.26
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.23/72.26
F(mark(z0)) → c11(F(z0)) 212.23/72.26
F(ok(z0)) → c12(F(z0)) 212.23/72.26
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.23/72.26
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.23/72.26
G(mark(z0)) → c15(G(z0)) 212.23/72.26
G(ok(z0)) → c16(G(z0)) 212.23/72.26
S(mark(z0)) → c17(S(z0)) 212.23/72.26
S(ok(z0)) → c18(S(z0)) 212.23/72.26
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.23/72.26
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.23/72.26
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.23/72.26
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0)) 212.23/72.26
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.23/72.26
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0)) 212.23/72.26
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.23/72.26
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.23/72.26
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.23/72.26
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.23/72.26
ACTIVE(f(z0)) → c(G(z0)) 212.23/72.26
ACTIVE(g(0)) → c1 212.23/72.26
ACTIVE(g(s(z0))) → c2(G(z0)) 212.23/72.26
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.23/72.26
ACTIVE(f(g(0))) → c5(F(mark(s(0))), ACTIVE(g(0))) 212.23/72.26
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.23/72.26
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.23/72.26
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.23/72.26
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.23/72.26
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.23/72.26
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.23/72.26
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.23/72.26
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.23/72.26
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.23/72.26
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.23/72.26
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1), ACTIVE(g(0))) 212.23/72.26
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.23/72.26
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.23/72.26
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.23/72.26
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.23/72.26
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.23/72.26
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.23/72.26
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.65/72.36
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.65/72.36
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1)))

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

212.65/72.36

(15) 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))) 212.65/72.36
ACTIVE(g(g(0))) → c7(G(mark(s(0))), ACTIVE(g(0))) 212.65/72.36
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.36
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.36
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.36
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.65/72.36
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.36
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.65/72.36
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.65/72.36
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.36
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))

212.65/72.36

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0)))) 212.65/72.36
active(g(0)) → mark(s(0)) 212.65/72.36
active(g(s(z0))) → mark(s(s(g(z0)))) 212.65/72.36
active(sel(0, cons(z0, z1))) → mark(z0) 212.65/72.36
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 212.65/72.36
active(f(z0)) → f(active(z0)) 212.65/72.36
active(cons(z0, z1)) → cons(active(z0), z1) 212.65/72.36
active(g(z0)) → g(active(z0)) 212.65/72.36
active(s(z0)) → s(active(z0)) 212.65/72.36
active(sel(z0, z1)) → sel(active(z0), z1) 212.65/72.36
active(sel(z0, z1)) → sel(z0, active(z1)) 212.65/72.36
f(mark(z0)) → mark(f(z0)) 212.65/72.36
f(ok(z0)) → ok(f(z0)) 212.65/72.36
cons(mark(z0), z1) → mark(cons(z0, z1)) 212.65/72.36
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 212.65/72.36
g(mark(z0)) → mark(g(z0)) 212.65/72.36
g(ok(z0)) → ok(g(z0)) 212.65/72.36
s(mark(z0)) → mark(s(z0)) 212.65/72.36
s(ok(z0)) → ok(s(z0)) 212.65/72.36
sel(mark(z0), z1) → mark(sel(z0, z1)) 212.65/72.36
sel(z0, mark(z1)) → mark(sel(z0, z1)) 212.65/72.36
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 212.65/72.36
proper(f(z0)) → f(proper(z0)) 212.65/72.36
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 212.65/72.36
proper(g(z0)) → g(proper(z0)) 212.65/72.36
proper(0) → ok(0) 212.65/72.36
proper(s(z0)) → s(proper(z0)) 212.65/72.36
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 212.65/72.36
top(mark(z0)) → top(proper(z0)) 212.65/72.36
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.65/72.36
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0)) 212.65/72.36
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.65/72.36
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.65/72.36
F(mark(z0)) → c11(F(z0)) 212.65/72.36
F(ok(z0)) → c12(F(z0)) 212.65/72.36
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.65/72.36
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.65/72.36
G(mark(z0)) → c15(G(z0)) 212.65/72.36
G(ok(z0)) → c16(G(z0)) 212.65/72.36
S(mark(z0)) → c17(S(z0)) 212.65/72.36
S(ok(z0)) → c18(S(z0)) 212.65/72.36
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.65/72.36
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.65/72.36
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.65/72.36
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0)) 212.65/72.36
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.65/72.36
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0)) 212.65/72.36
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.65/72.36
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.65/72.36
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.65/72.36
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.65/72.36
ACTIVE(f(z0)) → c(G(z0)) 212.65/72.36
ACTIVE(g(0)) → c1 212.65/72.36
ACTIVE(g(s(z0))) → c2(G(z0)) 212.65/72.36
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.36
ACTIVE(f(g(0))) → c5(F(mark(s(0))), ACTIVE(g(0))) 212.65/72.36
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.36
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.36
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.36
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.65/72.36
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.36
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.65/72.36
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.65/72.36
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.36
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.36
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.65/72.36
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1), ACTIVE(g(0))) 212.65/72.36
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.65/72.36
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.36
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.36
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.65/72.36
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.65/72.36
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.65/72.36
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.65/72.36
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.65/72.36
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.65/72.36
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.36
ACTIVE(g(g(0))) → c7(G(mark(s(0))), ACTIVE(g(0))) 212.65/72.36
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.36
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.36
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.36
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.65/72.36
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.36
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.65/72.36
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.65/72.36
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.36
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)) 212.65/72.36
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0)) 212.65/72.36
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.65/72.36
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.65/72.36
F(mark(z0)) → c11(F(z0)) 212.65/72.36
F(ok(z0)) → c12(F(z0)) 212.65/72.36
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.65/72.36
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.65/72.36
G(mark(z0)) → c15(G(z0)) 212.65/72.36
G(ok(z0)) → c16(G(z0)) 212.65/72.36
S(mark(z0)) → c17(S(z0)) 212.65/72.36
S(ok(z0)) → c18(S(z0)) 212.65/72.36
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.65/72.36
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.65/72.36
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.65/72.36
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0)) 212.65/72.36
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.65/72.36
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0)) 212.65/72.36
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.65/72.36
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.65/72.36
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.65/72.36
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.65/72.36
ACTIVE(f(z0)) → c(G(z0)) 212.65/72.36
ACTIVE(g(0)) → c1 212.65/72.36
ACTIVE(g(s(z0))) → c2(G(z0)) 212.65/72.36
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.36
ACTIVE(f(g(0))) → c5(F(mark(s(0))), ACTIVE(g(0))) 212.65/72.36
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.36
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.36
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.36
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.65/72.36
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.36
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.65/72.36
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.65/72.36
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.36
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.36
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.65/72.36
ACTIVE(cons(g(0), x1)) → c6(CONS(mark(s(0)), x1), ACTIVE(g(0))) 212.65/72.36
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.65/72.36
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.36
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.36
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.65/72.36
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.65/72.36
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.65/72.36
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.65/72.36
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.65/72.36
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.65/72.36
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.36
ACTIVE(g(g(0))) → c7(G(mark(s(0))), ACTIVE(g(0))) 212.65/72.36
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.36
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.36
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.36
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.65/72.36
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.36
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.65/72.36
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.65/72.36
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.36
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))

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

212.65/72.36

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

Split RHS of tuples not part of any SCC

212.65/72.36

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0)))) 212.65/72.36
active(g(0)) → mark(s(0)) 212.65/72.36
active(g(s(z0))) → mark(s(s(g(z0)))) 212.65/72.36
active(sel(0, cons(z0, z1))) → mark(z0) 212.65/72.36
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 212.65/72.36
active(f(z0)) → f(active(z0)) 212.65/72.36
active(cons(z0, z1)) → cons(active(z0), z1) 212.65/72.36
active(g(z0)) → g(active(z0)) 212.65/72.36
active(s(z0)) → s(active(z0)) 212.65/72.36
active(sel(z0, z1)) → sel(active(z0), z1) 212.65/72.36
active(sel(z0, z1)) → sel(z0, active(z1)) 212.65/72.36
f(mark(z0)) → mark(f(z0)) 212.65/72.36
f(ok(z0)) → ok(f(z0)) 212.65/72.36
cons(mark(z0), z1) → mark(cons(z0, z1)) 212.65/72.36
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 212.65/72.36
g(mark(z0)) → mark(g(z0)) 212.65/72.36
g(ok(z0)) → ok(g(z0)) 212.65/72.36
s(mark(z0)) → mark(s(z0)) 212.65/72.36
s(ok(z0)) → ok(s(z0)) 212.65/72.36
sel(mark(z0), z1) → mark(sel(z0, z1)) 212.65/72.36
sel(z0, mark(z1)) → mark(sel(z0, z1)) 212.65/72.36
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 212.65/72.38
proper(f(z0)) → f(proper(z0)) 212.65/72.38
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 212.65/72.38
proper(g(z0)) → g(proper(z0)) 212.65/72.38
proper(0) → ok(0) 212.65/72.38
proper(s(z0)) → s(proper(z0)) 212.65/72.38
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 212.65/72.38
top(mark(z0)) → top(proper(z0)) 212.65/72.38
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.65/72.38
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0)) 212.65/72.38
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.65/72.38
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.65/72.38
F(mark(z0)) → c11(F(z0)) 212.65/72.38
F(ok(z0)) → c12(F(z0)) 212.65/72.38
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.65/72.38
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.65/72.38
G(mark(z0)) → c15(G(z0)) 212.65/72.38
G(ok(z0)) → c16(G(z0)) 212.65/72.38
S(mark(z0)) → c17(S(z0)) 212.65/72.38
S(ok(z0)) → c18(S(z0)) 212.65/72.38
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.65/72.38
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.65/72.38
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.65/72.38
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0)) 212.65/72.38
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.65/72.38
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0)) 212.65/72.38
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.65/72.38
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.65/72.38
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.65/72.38
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.65/72.38
ACTIVE(f(z0)) → c(G(z0)) 212.65/72.38
ACTIVE(g(0)) → c1 212.65/72.38
ACTIVE(g(s(z0))) → c2(G(z0)) 212.65/72.38
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.38
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.38
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.38
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.38
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.65/72.38
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.65/72.38
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.65/72.38
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.38
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.38
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.65/72.38
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.65/72.38
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.65/72.38
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.38
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.38
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.38
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.38
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.65/72.38
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.65/72.38
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.65/72.38
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.65/72.38
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.65/72.38
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.65/72.38
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.65/72.38
ACTIVE(g(g(0))) → c3(ACTIVE(g(0)))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.65/72.38
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0)) 212.65/72.38
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.65/72.38
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.65/72.38
F(mark(z0)) → c11(F(z0)) 212.65/72.38
F(ok(z0)) → c12(F(z0)) 212.65/72.38
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.65/72.38
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.65/72.38
G(mark(z0)) → c15(G(z0)) 212.65/72.38
G(ok(z0)) → c16(G(z0)) 212.65/72.38
S(mark(z0)) → c17(S(z0)) 212.65/72.38
S(ok(z0)) → c18(S(z0)) 212.65/72.38
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.65/72.38
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.65/72.38
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.65/72.38
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0)) 212.65/72.38
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.65/72.38
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0)) 212.65/72.38
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.65/72.38
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.65/72.38
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.65/72.38
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.65/72.38
ACTIVE(f(z0)) → c(G(z0)) 212.65/72.38
ACTIVE(g(0)) → c1 212.65/72.38
ACTIVE(g(s(z0))) → c2(G(z0)) 212.65/72.38
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.38
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.38
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.38
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.38
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.65/72.38
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.65/72.38
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.65/72.38
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.38
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.38
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.65/72.38
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.65/72.38
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.65/72.38
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.38
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.38
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.38
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.38
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.65/72.38
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.65/72.38
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.65/72.38
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.65/72.38
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.65/72.38
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.65/72.38
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.65/72.38
ACTIVE(g(g(0))) → c3(ACTIVE(g(0)))

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

212.65/72.38

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

Removed 4 trailing nodes:

ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.65/72.38
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.65/72.38
ACTIVE(g(0)) → c1 212.65/72.38
ACTIVE(f(g(0))) → c3(ACTIVE(g(0)))

212.65/72.38

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0)))) 212.65/72.38
active(g(0)) → mark(s(0)) 212.65/72.38
active(g(s(z0))) → mark(s(s(g(z0)))) 212.65/72.38
active(sel(0, cons(z0, z1))) → mark(z0) 212.65/72.38
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 212.65/72.38
active(f(z0)) → f(active(z0)) 212.65/72.38
active(cons(z0, z1)) → cons(active(z0), z1) 212.65/72.38
active(g(z0)) → g(active(z0)) 212.65/72.38
active(s(z0)) → s(active(z0)) 212.65/72.38
active(sel(z0, z1)) → sel(active(z0), z1) 212.65/72.38
active(sel(z0, z1)) → sel(z0, active(z1)) 212.65/72.38
f(mark(z0)) → mark(f(z0)) 212.65/72.38
f(ok(z0)) → ok(f(z0)) 212.65/72.38
cons(mark(z0), z1) → mark(cons(z0, z1)) 212.65/72.38
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 212.65/72.38
g(mark(z0)) → mark(g(z0)) 212.65/72.38
g(ok(z0)) → ok(g(z0)) 212.65/72.38
s(mark(z0)) → mark(s(z0)) 212.65/72.38
s(ok(z0)) → ok(s(z0)) 212.65/72.38
sel(mark(z0), z1) → mark(sel(z0, z1)) 212.65/72.38
sel(z0, mark(z1)) → mark(sel(z0, z1)) 212.65/72.38
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 212.65/72.38
proper(f(z0)) → f(proper(z0)) 212.65/72.38
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 212.65/72.38
proper(g(z0)) → g(proper(z0)) 212.65/72.38
proper(0) → ok(0) 212.65/72.38
proper(s(z0)) → s(proper(z0)) 212.65/72.38
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 212.65/72.38
top(mark(z0)) → top(proper(z0)) 212.65/72.38
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.65/72.38
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0)) 212.65/72.38
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.65/72.38
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.65/72.38
F(mark(z0)) → c11(F(z0)) 212.65/72.38
F(ok(z0)) → c12(F(z0)) 212.65/72.38
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.65/72.38
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.65/72.38
G(mark(z0)) → c15(G(z0)) 212.65/72.38
G(ok(z0)) → c16(G(z0)) 212.65/72.38
S(mark(z0)) → c17(S(z0)) 212.65/72.38
S(ok(z0)) → c18(S(z0)) 212.65/72.38
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.65/72.38
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.65/72.38
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.65/72.38
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0)) 212.65/72.38
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.65/72.38
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0)) 212.65/72.38
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.65/72.38
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.65/72.38
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.65/72.38
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.65/72.38
ACTIVE(f(z0)) → c(G(z0)) 212.65/72.38
ACTIVE(g(0)) → c1 212.65/72.38
ACTIVE(g(s(z0))) → c2(G(z0)) 212.65/72.38
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.38
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.38
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.38
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.38
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.65/72.38
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.65/72.38
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.65/72.38
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.38
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.38
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.65/72.38
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.65/72.38
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.65/72.38
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.38
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.38
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.38
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.38
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.65/72.38
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.65/72.38
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.65/72.38
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.65/72.38
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.65/72.38
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.65/72.38
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.65/72.38
ACTIVE(g(g(0))) → c3(ACTIVE(g(0)))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.65/72.38
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0)) 212.65/72.38
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.65/72.38
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.65/72.38
F(mark(z0)) → c11(F(z0)) 212.65/72.38
F(ok(z0)) → c12(F(z0)) 212.65/72.38
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.65/72.38
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.65/72.38
G(mark(z0)) → c15(G(z0)) 212.65/72.38
G(ok(z0)) → c16(G(z0)) 212.65/72.38
S(mark(z0)) → c17(S(z0)) 212.65/72.38
S(ok(z0)) → c18(S(z0)) 212.65/72.38
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.65/72.38
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.65/72.38
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.65/72.38
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0)) 212.65/72.38
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.65/72.38
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0)) 212.65/72.38
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.65/72.38
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.65/72.38
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.65/72.38
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.65/72.38
ACTIVE(f(z0)) → c(G(z0)) 212.65/72.38
ACTIVE(g(0)) → c1 212.65/72.38
ACTIVE(g(s(z0))) → c2(G(z0)) 212.65/72.38
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.38
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.38
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.38
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.38
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.65/72.38
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.65/72.38
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.65/72.38
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.38
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.38
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.65/72.38
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.65/72.38
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.65/72.38
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.38
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.38
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.38
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.38
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.65/72.38
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.65/72.38
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.65/72.38
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.65/72.38
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.65/72.38
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.65/72.38
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.65/72.38
ACTIVE(g(g(0))) → c3(ACTIVE(g(0)))

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

212.65/72.38

(21) 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))) 212.65/72.38
ACTIVE(s(g(0))) → c8(S(mark(s(0))), ACTIVE(g(0))) 212.65/72.38
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.38
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.38
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.38
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.38
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.65/72.38
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.65/72.38
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))

212.65/72.38

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0)))) 212.65/72.38
active(g(0)) → mark(s(0)) 212.65/72.38
active(g(s(z0))) → mark(s(s(g(z0)))) 212.65/72.38
active(sel(0, cons(z0, z1))) → mark(z0) 212.65/72.38
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 212.65/72.38
active(f(z0)) → f(active(z0)) 212.65/72.38
active(cons(z0, z1)) → cons(active(z0), z1) 212.65/72.38
active(g(z0)) → g(active(z0)) 212.65/72.38
active(s(z0)) → s(active(z0)) 212.65/72.38
active(sel(z0, z1)) → sel(active(z0), z1) 212.65/72.38
active(sel(z0, z1)) → sel(z0, active(z1)) 212.65/72.38
f(mark(z0)) → mark(f(z0)) 212.65/72.38
f(ok(z0)) → ok(f(z0)) 212.65/72.38
cons(mark(z0), z1) → mark(cons(z0, z1)) 212.65/72.38
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 212.65/72.38
g(mark(z0)) → mark(g(z0)) 212.65/72.38
g(ok(z0)) → ok(g(z0)) 212.65/72.38
s(mark(z0)) → mark(s(z0)) 212.65/72.38
s(ok(z0)) → ok(s(z0)) 212.65/72.38
sel(mark(z0), z1) → mark(sel(z0, z1)) 212.65/72.38
sel(z0, mark(z1)) → mark(sel(z0, z1)) 212.65/72.38
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 212.65/72.38
proper(f(z0)) → f(proper(z0)) 212.65/72.38
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 212.65/72.38
proper(g(z0)) → g(proper(z0)) 212.65/72.38
proper(0) → ok(0) 212.65/72.38
proper(s(z0)) → s(proper(z0)) 212.65/72.38
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 212.65/72.38
top(mark(z0)) → top(proper(z0)) 212.65/72.38
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.65/72.38
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.65/72.38
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.65/72.38
F(mark(z0)) → c11(F(z0)) 212.65/72.38
F(ok(z0)) → c12(F(z0)) 212.65/72.38
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.65/72.38
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.65/72.38
G(mark(z0)) → c15(G(z0)) 212.65/72.38
G(ok(z0)) → c16(G(z0)) 212.65/72.38
S(mark(z0)) → c17(S(z0)) 212.65/72.38
S(ok(z0)) → c18(S(z0)) 212.65/72.38
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.65/72.38
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.65/72.38
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.65/72.38
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0)) 212.65/72.38
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.65/72.38
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0)) 212.65/72.38
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.65/72.38
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.65/72.38
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.65/72.38
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.65/72.38
ACTIVE(f(z0)) → c(G(z0)) 212.65/72.38
ACTIVE(g(0)) → c1 212.65/72.38
ACTIVE(g(s(z0))) → c2(G(z0)) 212.65/72.38
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.38
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.38
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.38
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.38
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.65/72.38
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.65/72.38
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.65/72.38
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.38
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.38
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.65/72.38
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.65/72.38
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.65/72.38
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.38
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.38
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.38
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.38
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.65/72.38
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.65/72.38
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.65/72.38
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.65/72.38
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.65/72.38
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.65/72.38
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.65/72.38
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.65/72.38
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(s(g(0))) → c8(S(mark(s(0))), ACTIVE(g(0))) 212.65/72.38
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.38
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.38
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.38
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.38
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.65/72.38
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.65/72.38
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.38
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)) 212.65/72.38
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.65/72.38
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.65/72.38
F(mark(z0)) → c11(F(z0)) 212.65/72.38
F(ok(z0)) → c12(F(z0)) 212.65/72.38
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.65/72.38
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.65/72.38
G(mark(z0)) → c15(G(z0)) 212.65/72.38
G(ok(z0)) → c16(G(z0)) 212.65/72.38
S(mark(z0)) → c17(S(z0)) 212.65/72.38
S(ok(z0)) → c18(S(z0)) 212.65/72.38
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.65/72.38
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.65/72.38
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.65/72.38
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0)) 212.65/72.38
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.65/72.38
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0)) 212.65/72.38
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.65/72.38
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.65/72.38
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.65/72.38
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.65/72.38
ACTIVE(f(z0)) → c(G(z0)) 212.65/72.38
ACTIVE(g(0)) → c1 212.65/72.38
ACTIVE(g(s(z0))) → c2(G(z0)) 212.65/72.38
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.38
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.38
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.38
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.38
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.65/72.38
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.65/72.38
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.65/72.38
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.38
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.38
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.65/72.38
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.65/72.38
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.65/72.38
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.38
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.38
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.38
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.38
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.65/72.38
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.65/72.38
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.65/72.38
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.65/72.38
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.65/72.38
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.65/72.38
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.65/72.38
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.65/72.38
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(s(g(0))) → c8(S(mark(s(0))), ACTIVE(g(0))) 212.65/72.38
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.38
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.38
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.38
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.38
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.65/72.38
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.65/72.38
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))

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

212.65/72.38

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

Split RHS of tuples not part of any SCC

212.65/72.38

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0)))) 212.65/72.38
active(g(0)) → mark(s(0)) 212.65/72.38
active(g(s(z0))) → mark(s(s(g(z0)))) 212.65/72.38
active(sel(0, cons(z0, z1))) → mark(z0) 212.65/72.38
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 212.65/72.38
active(f(z0)) → f(active(z0)) 212.65/72.38
active(cons(z0, z1)) → cons(active(z0), z1) 212.65/72.38
active(g(z0)) → g(active(z0)) 212.65/72.38
active(s(z0)) → s(active(z0)) 212.65/72.38
active(sel(z0, z1)) → sel(active(z0), z1) 212.65/72.38
active(sel(z0, z1)) → sel(z0, active(z1)) 212.65/72.38
f(mark(z0)) → mark(f(z0)) 212.65/72.38
f(ok(z0)) → ok(f(z0)) 212.65/72.38
cons(mark(z0), z1) → mark(cons(z0, z1)) 212.65/72.38
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 212.65/72.38
g(mark(z0)) → mark(g(z0)) 212.65/72.38
g(ok(z0)) → ok(g(z0)) 212.65/72.38
s(mark(z0)) → mark(s(z0)) 212.65/72.38
s(ok(z0)) → ok(s(z0)) 212.65/72.38
sel(mark(z0), z1) → mark(sel(z0, z1)) 212.65/72.38
sel(z0, mark(z1)) → mark(sel(z0, z1)) 212.65/72.38
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 212.65/72.38
proper(f(z0)) → f(proper(z0)) 212.65/72.38
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 212.65/72.38
proper(g(z0)) → g(proper(z0)) 212.65/72.38
proper(0) → ok(0) 212.65/72.38
proper(s(z0)) → s(proper(z0)) 212.65/72.38
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 212.65/72.38
top(mark(z0)) → top(proper(z0)) 212.65/72.38
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.65/72.38
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.65/72.38
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.65/72.38
F(mark(z0)) → c11(F(z0)) 212.65/72.38
F(ok(z0)) → c12(F(z0)) 212.65/72.38
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.65/72.38
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.65/72.38
G(mark(z0)) → c15(G(z0)) 212.65/72.38
G(ok(z0)) → c16(G(z0)) 212.65/72.38
S(mark(z0)) → c17(S(z0)) 212.65/72.38
S(ok(z0)) → c18(S(z0)) 212.65/72.38
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.65/72.38
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.65/72.38
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.65/72.38
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0)) 212.65/72.38
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.65/72.38
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0)) 212.65/72.38
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.65/72.38
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.65/72.38
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.65/72.38
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.65/72.38
ACTIVE(f(z0)) → c(G(z0)) 212.65/72.38
ACTIVE(g(0)) → c1 212.65/72.38
ACTIVE(g(s(z0))) → c2(G(z0)) 212.65/72.38
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.38
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.38
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.38
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.38
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.65/72.38
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.65/72.38
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.65/72.38
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.38
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.38
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.65/72.38
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.65/72.38
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.65/72.38
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.38
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.38
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.38
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.38
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.65/72.38
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.65/72.38
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.65/72.38
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.65/72.38
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.65/72.38
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.65/72.38
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.65/72.38
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.65/72.38
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.38
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.38
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.38
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.38
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.65/72.38
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.65/72.38
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.65/72.38
ACTIVE(s(g(0))) → c25(ACTIVE(g(0)))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.65/72.38
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.65/72.38
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.65/72.38
F(mark(z0)) → c11(F(z0)) 212.65/72.38
F(ok(z0)) → c12(F(z0)) 212.65/72.38
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.65/72.38
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.65/72.38
G(mark(z0)) → c15(G(z0)) 212.65/72.38
G(ok(z0)) → c16(G(z0)) 212.65/72.38
S(mark(z0)) → c17(S(z0)) 212.65/72.38
S(ok(z0)) → c18(S(z0)) 212.65/72.38
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.65/72.38
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.65/72.38
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.65/72.38
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0)) 212.65/72.38
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.65/72.38
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0)) 212.65/72.38
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.65/72.38
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.65/72.38
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.65/72.38
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.65/72.38
ACTIVE(f(z0)) → c(G(z0)) 212.65/72.38
ACTIVE(g(0)) → c1 212.65/72.38
ACTIVE(g(s(z0))) → c2(G(z0)) 212.65/72.38
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.38
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.38
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.38
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.38
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.65/72.38
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.65/72.38
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.65/72.38
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.38
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.38
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.65/72.38
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.65/72.38
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.65/72.38
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.38
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.38
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.38
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.38
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.65/72.38
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.65/72.38
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.65/72.38
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.65/72.38
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.65/72.38
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.65/72.38
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.65/72.38
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.65/72.38
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.38
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.38
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.38
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.38
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.65/72.38
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.65/72.38
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.65/72.38
ACTIVE(s(g(0))) → c25(ACTIVE(g(0)))

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

212.65/72.38

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

Removed 5 trailing nodes:

ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.65/72.38
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.65/72.38
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.65/72.38
ACTIVE(g(0)) → c1 212.65/72.38
ACTIVE(f(g(0))) → c3(ACTIVE(g(0)))

212.65/72.38

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0)))) 212.65/72.38
active(g(0)) → mark(s(0)) 212.65/72.38
active(g(s(z0))) → mark(s(s(g(z0)))) 212.65/72.38
active(sel(0, cons(z0, z1))) → mark(z0) 212.65/72.38
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 212.65/72.38
active(f(z0)) → f(active(z0)) 212.65/72.38
active(cons(z0, z1)) → cons(active(z0), z1) 212.65/72.38
active(g(z0)) → g(active(z0)) 212.65/72.38
active(s(z0)) → s(active(z0)) 212.65/72.38
active(sel(z0, z1)) → sel(active(z0), z1) 212.65/72.38
active(sel(z0, z1)) → sel(z0, active(z1)) 212.65/72.38
f(mark(z0)) → mark(f(z0)) 212.65/72.38
f(ok(z0)) → ok(f(z0)) 212.65/72.38
cons(mark(z0), z1) → mark(cons(z0, z1)) 212.65/72.38
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 212.65/72.38
g(mark(z0)) → mark(g(z0)) 212.65/72.38
g(ok(z0)) → ok(g(z0)) 212.65/72.38
s(mark(z0)) → mark(s(z0)) 212.65/72.38
s(ok(z0)) → ok(s(z0)) 212.65/72.38
sel(mark(z0), z1) → mark(sel(z0, z1)) 212.65/72.38
sel(z0, mark(z1)) → mark(sel(z0, z1)) 212.65/72.38
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 212.65/72.38
proper(f(z0)) → f(proper(z0)) 212.65/72.38
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 212.65/72.38
proper(g(z0)) → g(proper(z0)) 212.65/72.38
proper(0) → ok(0) 212.65/72.38
proper(s(z0)) → s(proper(z0)) 212.65/72.38
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 212.65/72.38
top(mark(z0)) → top(proper(z0)) 212.65/72.38
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.65/72.38
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.65/72.38
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.65/72.38
F(mark(z0)) → c11(F(z0)) 212.65/72.38
F(ok(z0)) → c12(F(z0)) 212.65/72.38
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.65/72.38
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.65/72.38
G(mark(z0)) → c15(G(z0)) 212.65/72.38
G(ok(z0)) → c16(G(z0)) 212.65/72.38
S(mark(z0)) → c17(S(z0)) 212.65/72.38
S(ok(z0)) → c18(S(z0)) 212.65/72.38
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.65/72.38
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.65/72.38
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.65/72.38
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0)) 212.65/72.38
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.65/72.38
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0)) 212.65/72.38
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.65/72.38
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.65/72.38
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.65/72.38
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.65/72.38
ACTIVE(f(z0)) → c(G(z0)) 212.65/72.38
ACTIVE(g(0)) → c1 212.65/72.38
ACTIVE(g(s(z0))) → c2(G(z0)) 212.65/72.38
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.38
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.38
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.38
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.38
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.65/72.38
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.65/72.38
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.65/72.38
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.38
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.38
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.65/72.38
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.65/72.38
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.65/72.38
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.38
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.38
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.38
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.38
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.65/72.38
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.65/72.38
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.65/72.38
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.65/72.38
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.65/72.38
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.65/72.38
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.65/72.38
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.65/72.38
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.38
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.38
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.38
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.38
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.65/72.38
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.65/72.38
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.65/72.38
ACTIVE(s(g(0))) → c25(ACTIVE(g(0)))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.65/72.38
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.65/72.38
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.65/72.38
F(mark(z0)) → c11(F(z0)) 212.65/72.38
F(ok(z0)) → c12(F(z0)) 212.65/72.38
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.65/72.38
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.65/72.38
G(mark(z0)) → c15(G(z0)) 212.65/72.38
G(ok(z0)) → c16(G(z0)) 212.65/72.38
S(mark(z0)) → c17(S(z0)) 212.65/72.38
S(ok(z0)) → c18(S(z0)) 212.65/72.38
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.65/72.38
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.65/72.38
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.65/72.38
PROPER(f(z0)) → c22(F(proper(z0)), PROPER(z0)) 212.65/72.38
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.65/72.38
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0)) 212.65/72.38
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.65/72.38
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.65/72.38
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.65/72.38
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.65/72.38
ACTIVE(f(z0)) → c(G(z0)) 212.65/72.38
ACTIVE(g(0)) → c1 212.65/72.38
ACTIVE(g(s(z0))) → c2(G(z0)) 212.65/72.38
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.38
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.38
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.38
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.38
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.65/72.38
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.65/72.38
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.65/72.38
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.38
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.38
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.65/72.38
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.65/72.38
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.65/72.38
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.38
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.38
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.38
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.38
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.65/72.38
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.65/72.38
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.65/72.38
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.65/72.38
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.65/72.38
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.65/72.38
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.65/72.38
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.65/72.38
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.38
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.38
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.38
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.38
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.65/72.38
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.65/72.38
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.65/72.38
ACTIVE(s(g(0))) → c25(ACTIVE(g(0)))

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

212.65/72.38

(27) 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))) 212.65/72.38
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.65/72.38
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0))) 212.65/72.38
PROPER(f(0)) → c22(F(ok(0)), PROPER(0)) 212.65/72.38
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0))) 212.65/72.38
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))

212.65/72.38

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0)))) 212.65/72.38
active(g(0)) → mark(s(0)) 212.65/72.38
active(g(s(z0))) → mark(s(s(g(z0)))) 212.65/72.38
active(sel(0, cons(z0, z1))) → mark(z0) 212.65/72.38
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 212.65/72.38
active(f(z0)) → f(active(z0)) 212.65/72.38
active(cons(z0, z1)) → cons(active(z0), z1) 212.65/72.38
active(g(z0)) → g(active(z0)) 212.65/72.38
active(s(z0)) → s(active(z0)) 212.65/72.38
active(sel(z0, z1)) → sel(active(z0), z1) 212.65/72.38
active(sel(z0, z1)) → sel(z0, active(z1)) 212.65/72.38
f(mark(z0)) → mark(f(z0)) 212.65/72.38
f(ok(z0)) → ok(f(z0)) 212.65/72.38
cons(mark(z0), z1) → mark(cons(z0, z1)) 212.65/72.38
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 212.65/72.38
g(mark(z0)) → mark(g(z0)) 212.65/72.38
g(ok(z0)) → ok(g(z0)) 212.65/72.38
s(mark(z0)) → mark(s(z0)) 212.65/72.38
s(ok(z0)) → ok(s(z0)) 212.65/72.38
sel(mark(z0), z1) → mark(sel(z0, z1)) 212.65/72.38
sel(z0, mark(z1)) → mark(sel(z0, z1)) 212.65/72.38
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 212.65/72.38
proper(f(z0)) → f(proper(z0)) 212.65/72.38
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 212.65/72.38
proper(g(z0)) → g(proper(z0)) 212.65/72.38
proper(0) → ok(0) 212.65/72.38
proper(s(z0)) → s(proper(z0)) 212.65/72.38
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 212.65/72.38
top(mark(z0)) → top(proper(z0)) 212.65/72.38
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.65/72.38
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.65/72.38
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.65/72.38
F(mark(z0)) → c11(F(z0)) 212.65/72.38
F(ok(z0)) → c12(F(z0)) 212.65/72.38
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.65/72.38
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.65/72.38
G(mark(z0)) → c15(G(z0)) 212.65/72.38
G(ok(z0)) → c16(G(z0)) 212.65/72.38
S(mark(z0)) → c17(S(z0)) 212.65/72.38
S(ok(z0)) → c18(S(z0)) 212.65/72.38
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.65/72.38
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.65/72.38
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.65/72.38
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.65/72.38
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0)) 212.65/72.38
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.65/72.38
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.65/72.38
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.65/72.38
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.65/72.38
ACTIVE(f(z0)) → c(G(z0)) 212.65/72.38
ACTIVE(g(0)) → c1 212.65/72.38
ACTIVE(g(s(z0))) → c2(G(z0)) 212.65/72.38
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.38
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.38
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.38
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.38
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.65/72.38
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.65/72.38
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.38
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.65/72.38
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.38
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.38
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.65/72.38
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.65/72.38
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.65/72.39
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.65/72.39
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.39
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.39
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.39
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.39
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.65/72.39
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.65/72.39
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.65/72.39
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.65/72.39
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.65/72.39
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.65/72.39
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.65/72.39
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.65/72.39
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.39
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.39
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.39
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.39
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.65/72.39
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.65/72.39
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.65/72.39
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.65/72.39
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0))) 212.65/72.39
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.65/72.39
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0))) 212.65/72.39
PROPER(f(0)) → c22(F(ok(0)), PROPER(0)) 212.65/72.39
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0))) 212.65/72.39
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)) 212.65/72.39
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.65/72.39
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.65/72.39
F(mark(z0)) → c11(F(z0)) 212.65/72.39
F(ok(z0)) → c12(F(z0)) 212.65/72.39
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.65/72.39
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.65/72.39
G(mark(z0)) → c15(G(z0)) 212.65/72.39
G(ok(z0)) → c16(G(z0)) 212.65/72.39
S(mark(z0)) → c17(S(z0)) 212.65/72.39
S(ok(z0)) → c18(S(z0)) 212.65/72.39
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.65/72.39
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.65/72.39
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.65/72.39
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.65/72.39
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0)) 212.65/72.39
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.65/72.39
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.65/72.39
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.65/72.39
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.65/72.39
ACTIVE(f(z0)) → c(G(z0)) 212.65/72.39
ACTIVE(g(0)) → c1 212.65/72.39
ACTIVE(g(s(z0))) → c2(G(z0)) 212.65/72.39
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.39
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.39
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.39
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.39
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.65/72.39
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.65/72.39
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.65/72.39
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.39
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.39
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.65/72.39
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.65/72.39
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.65/72.39
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.39
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.39
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.39
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.39
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.65/72.39
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.65/72.39
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.65/72.39
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.65/72.39
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.65/72.39
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.65/72.39
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.65/72.39
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.65/72.39
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.39
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.39
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.39
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.39
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.65/72.39
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.65/72.39
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.65/72.39
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.65/72.39
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0))) 212.65/72.39
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.65/72.39
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0))) 212.65/72.39
PROPER(f(0)) → c22(F(ok(0)), PROPER(0)) 212.65/72.39
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0))) 212.65/72.39
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))

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

212.65/72.39

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

Removed 1 trailing tuple parts

212.65/72.39

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0)))) 212.65/72.39
active(g(0)) → mark(s(0)) 212.65/72.39
active(g(s(z0))) → mark(s(s(g(z0)))) 212.65/72.39
active(sel(0, cons(z0, z1))) → mark(z0) 212.65/72.39
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 212.65/72.39
active(f(z0)) → f(active(z0)) 212.65/72.39
active(cons(z0, z1)) → cons(active(z0), z1) 212.65/72.39
active(g(z0)) → g(active(z0)) 212.65/72.39
active(s(z0)) → s(active(z0)) 212.65/72.39
active(sel(z0, z1)) → sel(active(z0), z1) 212.65/72.39
active(sel(z0, z1)) → sel(z0, active(z1)) 212.65/72.39
f(mark(z0)) → mark(f(z0)) 212.65/72.39
f(ok(z0)) → ok(f(z0)) 212.65/72.39
cons(mark(z0), z1) → mark(cons(z0, z1)) 212.65/72.39
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 212.65/72.39
g(mark(z0)) → mark(g(z0)) 212.65/72.39
g(ok(z0)) → ok(g(z0)) 212.65/72.39
s(mark(z0)) → mark(s(z0)) 212.65/72.39
s(ok(z0)) → ok(s(z0)) 212.65/72.39
sel(mark(z0), z1) → mark(sel(z0, z1)) 212.65/72.39
sel(z0, mark(z1)) → mark(sel(z0, z1)) 212.65/72.39
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 212.65/72.39
proper(f(z0)) → f(proper(z0)) 212.65/72.39
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 212.65/72.39
proper(g(z0)) → g(proper(z0)) 212.65/72.39
proper(0) → ok(0) 212.65/72.39
proper(s(z0)) → s(proper(z0)) 212.65/72.39
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 212.65/72.39
top(mark(z0)) → top(proper(z0)) 212.65/72.39
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.65/72.39
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.65/72.39
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.65/72.39
F(mark(z0)) → c11(F(z0)) 212.65/72.39
F(ok(z0)) → c12(F(z0)) 212.65/72.39
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.65/72.39
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.65/72.39
G(mark(z0)) → c15(G(z0)) 212.65/72.39
G(ok(z0)) → c16(G(z0)) 212.65/72.39
S(mark(z0)) → c17(S(z0)) 212.65/72.39
S(ok(z0)) → c18(S(z0)) 212.65/72.39
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.65/72.39
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.65/72.39
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.65/72.39
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.65/72.39
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0)) 212.65/72.39
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.65/72.39
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.65/72.39
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.65/72.39
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.65/72.39
ACTIVE(f(z0)) → c(G(z0)) 212.65/72.39
ACTIVE(g(0)) → c1 212.65/72.39
ACTIVE(g(s(z0))) → c2(G(z0)) 212.65/72.39
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.39
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.39
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.39
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.39
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.65/72.39
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.65/72.39
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.65/72.39
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.39
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.39
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.65/72.39
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.65/72.39
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.65/72.39
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.39
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.39
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.39
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.39
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.65/72.39
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.65/72.39
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.65/72.39
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.65/72.39
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.65/72.39
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.65/72.39
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.65/72.39
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.65/72.39
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.39
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.39
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.39
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.39
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.65/72.39
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.65/72.39
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.65/72.39
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.65/72.39
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0))) 212.65/72.39
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.65/72.39
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0))) 212.65/72.39
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0))) 212.65/72.39
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.65/72.39
PROPER(f(0)) → c22(F(ok(0)))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.65/72.39
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.65/72.39
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.65/72.39
F(mark(z0)) → c11(F(z0)) 212.65/72.39
F(ok(z0)) → c12(F(z0)) 212.65/72.39
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.65/72.39
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.65/72.39
G(mark(z0)) → c15(G(z0)) 212.65/72.39
G(ok(z0)) → c16(G(z0)) 212.65/72.39
S(mark(z0)) → c17(S(z0)) 212.65/72.39
S(ok(z0)) → c18(S(z0)) 212.65/72.39
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.65/72.39
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.65/72.39
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.65/72.39
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.65/72.39
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0)) 212.65/72.39
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.65/72.39
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.65/72.39
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.65/72.39
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.65/72.39
ACTIVE(f(z0)) → c(G(z0)) 212.65/72.39
ACTIVE(g(0)) → c1 212.65/72.39
ACTIVE(g(s(z0))) → c2(G(z0)) 212.65/72.39
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.39
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.39
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.39
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.39
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.65/72.39
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.65/72.39
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.65/72.39
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.39
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.39
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.65/72.39
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.65/72.39
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.65/72.39
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.39
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.39
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.39
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.39
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.65/72.39
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.65/72.39
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.65/72.39
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.65/72.39
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.65/72.39
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.65/72.39
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.65/72.39
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.65/72.39
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.39
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.39
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.39
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.39
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.65/72.39
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.65/72.39
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.65/72.39
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.65/72.39
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0))) 212.65/72.39
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.65/72.39
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0))) 212.65/72.39
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0))) 212.65/72.39
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.65/72.39
PROPER(f(0)) → c22(F(ok(0)))

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

212.65/72.39

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

Removed 5 trailing nodes:

ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.65/72.39
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.65/72.39
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.65/72.39
ACTIVE(g(0)) → c1 212.65/72.39
ACTIVE(f(g(0))) → c3(ACTIVE(g(0)))

212.65/72.39

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0)))) 212.65/72.39
active(g(0)) → mark(s(0)) 212.65/72.39
active(g(s(z0))) → mark(s(s(g(z0)))) 212.65/72.39
active(sel(0, cons(z0, z1))) → mark(z0) 212.65/72.39
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 212.65/72.39
active(f(z0)) → f(active(z0)) 212.65/72.39
active(cons(z0, z1)) → cons(active(z0), z1) 212.65/72.39
active(g(z0)) → g(active(z0)) 212.65/72.39
active(s(z0)) → s(active(z0)) 212.65/72.39
active(sel(z0, z1)) → sel(active(z0), z1) 212.65/72.39
active(sel(z0, z1)) → sel(z0, active(z1)) 212.65/72.39
f(mark(z0)) → mark(f(z0)) 212.65/72.39
f(ok(z0)) → ok(f(z0)) 212.65/72.39
cons(mark(z0), z1) → mark(cons(z0, z1)) 212.65/72.39
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 212.65/72.39
g(mark(z0)) → mark(g(z0)) 212.65/72.39
g(ok(z0)) → ok(g(z0)) 212.65/72.39
s(mark(z0)) → mark(s(z0)) 212.65/72.39
s(ok(z0)) → ok(s(z0)) 212.65/72.39
sel(mark(z0), z1) → mark(sel(z0, z1)) 212.65/72.39
sel(z0, mark(z1)) → mark(sel(z0, z1)) 212.65/72.39
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 212.65/72.39
proper(f(z0)) → f(proper(z0)) 212.65/72.39
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 212.65/72.39
proper(g(z0)) → g(proper(z0)) 212.65/72.39
proper(0) → ok(0) 212.65/72.39
proper(s(z0)) → s(proper(z0)) 212.65/72.39
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 212.65/72.39
top(mark(z0)) → top(proper(z0)) 212.65/72.39
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.65/72.39
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.65/72.39
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.65/72.39
F(mark(z0)) → c11(F(z0)) 212.65/72.39
F(ok(z0)) → c12(F(z0)) 212.65/72.39
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.65/72.39
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.65/72.39
G(mark(z0)) → c15(G(z0)) 212.65/72.39
G(ok(z0)) → c16(G(z0)) 212.65/72.39
S(mark(z0)) → c17(S(z0)) 212.65/72.39
S(ok(z0)) → c18(S(z0)) 212.65/72.39
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.65/72.39
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.65/72.39
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.65/72.39
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.65/72.39
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0)) 212.65/72.39
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.65/72.39
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.65/72.39
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.65/72.39
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.65/72.39
ACTIVE(f(z0)) → c(G(z0)) 212.65/72.39
ACTIVE(g(0)) → c1 212.65/72.39
ACTIVE(g(s(z0))) → c2(G(z0)) 212.65/72.39
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.39
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.39
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.39
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.39
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.65/72.39
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.65/72.39
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.65/72.39
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.39
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.39
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.65/72.39
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.65/72.39
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.65/72.39
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.39
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.39
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.39
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.39
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.65/72.39
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.65/72.39
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.65/72.39
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.65/72.39
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.65/72.39
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.65/72.39
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.65/72.39
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.65/72.39
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.39
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.39
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.39
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.39
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.65/72.39
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.65/72.39
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.65/72.39
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.65/72.39
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0))) 212.65/72.39
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.65/72.39
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0))) 212.65/72.39
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0))) 212.65/72.39
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.65/72.39
PROPER(f(0)) → c22(F(ok(0)))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.65/72.39
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.65/72.39
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.65/72.39
F(mark(z0)) → c11(F(z0)) 212.65/72.39
F(ok(z0)) → c12(F(z0)) 212.65/72.39
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.65/72.39
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.65/72.39
G(mark(z0)) → c15(G(z0)) 212.65/72.39
G(ok(z0)) → c16(G(z0)) 212.65/72.39
S(mark(z0)) → c17(S(z0)) 212.65/72.39
S(ok(z0)) → c18(S(z0)) 212.65/72.39
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.65/72.39
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.65/72.39
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.65/72.39
PROPER(cons(z0, z1)) → c23(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.65/72.39
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0)) 212.65/72.39
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.65/72.39
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.65/72.39
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.65/72.39
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.65/72.39
ACTIVE(f(z0)) → c(G(z0)) 212.65/72.39
ACTIVE(g(0)) → c1 212.65/72.39
ACTIVE(g(s(z0))) → c2(G(z0)) 212.65/72.39
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.39
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.39
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.39
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.39
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.65/72.39
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.65/72.39
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.65/72.39
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.39
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.39
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.65/72.39
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.65/72.39
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.65/72.39
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.39
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.39
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.39
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.39
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.65/72.39
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.65/72.39
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.65/72.39
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.65/72.39
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.65/72.39
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.65/72.39
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.65/72.39
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.65/72.39
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.39
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.39
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.39
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.39
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.65/72.39
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.65/72.39
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.65/72.39
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.65/72.39
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0))) 212.65/72.39
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.65/72.39
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0))) 212.65/72.39
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0))) 212.65/72.39
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.65/72.39
PROPER(f(0)) → c22(F(ok(0)))

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

212.65/72.39

(33) 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))) 212.65/72.39
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.65/72.39
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.65/72.39
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 212.65/72.39
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.65/72.39
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.65/72.39
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.65/72.39
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.65/72.39
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.65/72.39
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 212.65/72.39
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.65/72.39
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))

212.65/72.39

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0)))) 212.65/72.39
active(g(0)) → mark(s(0)) 212.65/72.39
active(g(s(z0))) → mark(s(s(g(z0)))) 212.65/72.39
active(sel(0, cons(z0, z1))) → mark(z0) 212.65/72.39
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 212.65/72.39
active(f(z0)) → f(active(z0)) 212.65/72.39
active(cons(z0, z1)) → cons(active(z0), z1) 212.65/72.39
active(g(z0)) → g(active(z0)) 212.65/72.39
active(s(z0)) → s(active(z0)) 212.65/72.39
active(sel(z0, z1)) → sel(active(z0), z1) 212.65/72.39
active(sel(z0, z1)) → sel(z0, active(z1)) 212.65/72.39
f(mark(z0)) → mark(f(z0)) 212.65/72.39
f(ok(z0)) → ok(f(z0)) 212.65/72.39
cons(mark(z0), z1) → mark(cons(z0, z1)) 212.65/72.39
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 212.65/72.39
g(mark(z0)) → mark(g(z0)) 212.65/72.39
g(ok(z0)) → ok(g(z0)) 212.65/72.39
s(mark(z0)) → mark(s(z0)) 212.65/72.39
s(ok(z0)) → ok(s(z0)) 212.65/72.39
sel(mark(z0), z1) → mark(sel(z0, z1)) 212.65/72.39
sel(z0, mark(z1)) → mark(sel(z0, z1)) 212.65/72.39
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 212.65/72.39
proper(f(z0)) → f(proper(z0)) 212.65/72.39
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 212.65/72.39
proper(g(z0)) → g(proper(z0)) 212.65/72.39
proper(0) → ok(0) 212.65/72.39
proper(s(z0)) → s(proper(z0)) 212.65/72.39
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 212.65/72.39
top(mark(z0)) → top(proper(z0)) 212.65/72.39
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.65/72.39
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.65/72.39
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.65/72.39
F(mark(z0)) → c11(F(z0)) 212.65/72.39
F(ok(z0)) → c12(F(z0)) 212.65/72.39
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.65/72.39
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.65/72.39
G(mark(z0)) → c15(G(z0)) 212.65/72.39
G(ok(z0)) → c16(G(z0)) 212.65/72.39
S(mark(z0)) → c17(S(z0)) 212.65/72.39
S(ok(z0)) → c18(S(z0)) 212.65/72.39
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.65/72.39
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.65/72.39
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.65/72.39
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0)) 212.65/72.39
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.65/72.39
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.65/72.39
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.65/72.39
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.65/72.39
ACTIVE(f(z0)) → c(G(z0)) 212.65/72.39
ACTIVE(g(0)) → c1 212.65/72.39
ACTIVE(g(s(z0))) → c2(G(z0)) 212.65/72.39
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.39
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.39
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.39
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.39
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.65/72.39
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.65/72.39
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.65/72.39
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.39
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.39
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.65/72.39
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.65/72.39
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.65/72.39
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.39
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.39
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.39
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.39
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.65/72.39
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.65/72.39
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.65/72.39
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.65/72.39
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.65/72.39
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.65/72.39
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.65/72.39
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.65/72.39
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.39
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.39
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.39
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.39
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.65/72.39
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.65/72.39
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.65/72.39
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.65/72.39
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0))) 212.65/72.39
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.65/72.39
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0))) 212.65/72.39
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0))) 212.65/72.39
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.65/72.39
PROPER(f(0)) → c22(F(ok(0))) 212.65/72.39
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.65/72.39
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.65/72.39
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.65/72.39
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 212.65/72.39
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.65/72.39
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.65/72.39
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.65/72.39
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.65/72.39
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.65/72.39
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 212.65/72.39
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.65/72.39
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)) 212.65/72.39
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.65/72.39
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.65/72.39
F(mark(z0)) → c11(F(z0)) 212.65/72.39
F(ok(z0)) → c12(F(z0)) 212.65/72.39
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.65/72.39
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.65/72.39
G(mark(z0)) → c15(G(z0)) 212.65/72.39
G(ok(z0)) → c16(G(z0)) 212.65/72.39
S(mark(z0)) → c17(S(z0)) 212.65/72.39
S(ok(z0)) → c18(S(z0)) 212.65/72.39
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.65/72.39
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.65/72.39
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.65/72.39
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0)) 212.65/72.39
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.65/72.39
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.65/72.39
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.65/72.39
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.65/72.39
ACTIVE(f(z0)) → c(G(z0)) 212.65/72.39
ACTIVE(g(0)) → c1 212.65/72.39
ACTIVE(g(s(z0))) → c2(G(z0)) 212.65/72.39
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.39
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.39
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.39
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.39
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.65/72.39
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.65/72.39
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.65/72.39
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.39
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.39
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.65/72.39
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.65/72.39
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.65/72.39
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.39
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.39
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.39
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.39
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.65/72.39
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.65/72.39
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.65/72.39
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.65/72.39
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.65/72.39
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.65/72.39
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.65/72.39
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.65/72.39
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.39
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.39
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.39
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.39
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.65/72.39
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.65/72.39
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.65/72.39
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.65/72.39
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0))) 212.65/72.39
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.65/72.39
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0))) 212.65/72.39
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0))) 212.65/72.39
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.65/72.39
PROPER(f(0)) → c22(F(ok(0))) 212.65/72.39
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.65/72.39
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.65/72.39
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.65/72.39
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 212.65/72.39
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.65/72.39
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.65/72.39
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.65/72.39
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.65/72.39
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.65/72.39
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 212.65/72.39
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.65/72.39
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, f, cons, g, s, sel, proper, top

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, c1, c2, c5, c6, c7, c3, c8, c25, c22, c22, c23

212.65/72.39

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

Removed 2 trailing tuple parts

212.65/72.39

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0)))) 212.65/72.39
active(g(0)) → mark(s(0)) 212.65/72.39
active(g(s(z0))) → mark(s(s(g(z0)))) 212.65/72.39
active(sel(0, cons(z0, z1))) → mark(z0) 212.65/72.39
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 212.65/72.39
active(f(z0)) → f(active(z0)) 212.65/72.39
active(cons(z0, z1)) → cons(active(z0), z1) 212.65/72.39
active(g(z0)) → g(active(z0)) 212.65/72.39
active(s(z0)) → s(active(z0)) 212.65/72.39
active(sel(z0, z1)) → sel(active(z0), z1) 212.65/72.39
active(sel(z0, z1)) → sel(z0, active(z1)) 212.65/72.39
f(mark(z0)) → mark(f(z0)) 212.65/72.39
f(ok(z0)) → ok(f(z0)) 212.65/72.39
cons(mark(z0), z1) → mark(cons(z0, z1)) 212.65/72.39
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 212.65/72.39
g(mark(z0)) → mark(g(z0)) 212.65/72.39
g(ok(z0)) → ok(g(z0)) 212.65/72.39
s(mark(z0)) → mark(s(z0)) 212.65/72.39
s(ok(z0)) → ok(s(z0)) 212.65/72.39
sel(mark(z0), z1) → mark(sel(z0, z1)) 212.65/72.39
sel(z0, mark(z1)) → mark(sel(z0, z1)) 212.65/72.39
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 212.65/72.39
proper(f(z0)) → f(proper(z0)) 212.65/72.39
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 212.65/72.39
proper(g(z0)) → g(proper(z0)) 212.65/72.39
proper(0) → ok(0) 212.65/72.39
proper(s(z0)) → s(proper(z0)) 212.65/72.39
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 212.65/72.39
top(mark(z0)) → top(proper(z0)) 212.65/72.39
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.65/72.39
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.65/72.39
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.65/72.39
F(mark(z0)) → c11(F(z0)) 212.65/72.39
F(ok(z0)) → c12(F(z0)) 212.65/72.39
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.65/72.39
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.65/72.39
G(mark(z0)) → c15(G(z0)) 212.65/72.39
G(ok(z0)) → c16(G(z0)) 212.65/72.39
S(mark(z0)) → c17(S(z0)) 212.65/72.39
S(ok(z0)) → c18(S(z0)) 212.65/72.39
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.65/72.39
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.65/72.39
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.65/72.39
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0)) 212.65/72.39
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.65/72.39
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.65/72.39
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.65/72.39
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.65/72.39
ACTIVE(f(z0)) → c(G(z0)) 212.65/72.39
ACTIVE(g(0)) → c1 212.65/72.39
ACTIVE(g(s(z0))) → c2(G(z0)) 212.65/72.39
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.39
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.39
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.39
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.39
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.65/72.39
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.65/72.39
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.65/72.39
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.39
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.39
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.65/72.39
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.65/72.39
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.65/72.39
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.39
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.39
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.39
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.39
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.65/72.39
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.65/72.39
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.65/72.39
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.65/72.39
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.65/72.39
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.65/72.39
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.65/72.39
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.65/72.39
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.39
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.39
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.39
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.39
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.65/72.39
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.65/72.39
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.65/72.39
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.65/72.39
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0))) 212.65/72.39
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.65/72.39
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0))) 212.65/72.39
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0))) 212.65/72.39
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.65/72.39
PROPER(f(0)) → c22(F(ok(0))) 212.65/72.39
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.65/72.39
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.65/72.39
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.65/72.39
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.65/72.39
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.65/72.39
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.65/72.39
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.65/72.39
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.65/72.39
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.65/72.39
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.65/72.39
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0)) 212.65/72.39
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)) 212.65/72.39
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.65/72.39
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.65/72.39
F(mark(z0)) → c11(F(z0)) 212.65/72.39
F(ok(z0)) → c12(F(z0)) 212.65/72.39
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.65/72.39
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.65/72.39
G(mark(z0)) → c15(G(z0)) 212.65/72.39
G(ok(z0)) → c16(G(z0)) 212.65/72.39
S(mark(z0)) → c17(S(z0)) 212.65/72.39
S(ok(z0)) → c18(S(z0)) 212.65/72.39
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.65/72.39
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.65/72.39
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.65/72.39
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0)) 212.65/72.39
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.65/72.39
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.65/72.39
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.65/72.39
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.65/72.39
ACTIVE(f(z0)) → c(G(z0)) 212.65/72.39
ACTIVE(g(0)) → c1 212.65/72.39
ACTIVE(g(s(z0))) → c2(G(z0)) 212.65/72.39
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.39
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.39
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.39
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.39
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.65/72.39
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.65/72.39
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.65/72.39
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.39
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.39
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.65/72.39
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.65/72.39
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.65/72.39
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.39
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.39
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.39
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.39
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.65/72.39
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.65/72.39
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.65/72.39
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.65/72.39
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.65/72.39
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.65/72.39
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.65/72.39
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.65/72.39
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.39
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.39
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.39
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.39
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.65/72.39
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.65/72.39
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.65/72.39
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.65/72.39
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0))) 212.65/72.39
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.65/72.39
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0))) 212.65/72.39
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0))) 212.65/72.39
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.65/72.39
PROPER(f(0)) → c22(F(ok(0))) 212.65/72.39
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.65/72.39
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.65/72.39
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.65/72.39
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.65/72.39
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.65/72.39
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.65/72.39
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.65/72.39
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.65/72.39
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.65/72.39
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.65/72.39
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0)) 212.65/72.39
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1))

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

212.65/72.39

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

Removed 5 trailing nodes:

ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.65/72.39
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.65/72.39
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.65/72.39
ACTIVE(g(0)) → c1 212.65/72.39
ACTIVE(f(g(0))) → c3(ACTIVE(g(0)))

212.65/72.39

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0)))) 212.65/72.39
active(g(0)) → mark(s(0)) 212.65/72.39
active(g(s(z0))) → mark(s(s(g(z0)))) 212.65/72.39
active(sel(0, cons(z0, z1))) → mark(z0) 212.65/72.39
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 212.65/72.39
active(f(z0)) → f(active(z0)) 212.65/72.39
active(cons(z0, z1)) → cons(active(z0), z1) 212.65/72.39
active(g(z0)) → g(active(z0)) 212.65/72.39
active(s(z0)) → s(active(z0)) 212.65/72.39
active(sel(z0, z1)) → sel(active(z0), z1) 212.65/72.39
active(sel(z0, z1)) → sel(z0, active(z1)) 212.65/72.39
f(mark(z0)) → mark(f(z0)) 212.65/72.39
f(ok(z0)) → ok(f(z0)) 212.65/72.39
cons(mark(z0), z1) → mark(cons(z0, z1)) 212.65/72.39
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 212.65/72.39
g(mark(z0)) → mark(g(z0)) 212.65/72.39
g(ok(z0)) → ok(g(z0)) 212.65/72.39
s(mark(z0)) → mark(s(z0)) 212.65/72.39
s(ok(z0)) → ok(s(z0)) 212.65/72.39
sel(mark(z0), z1) → mark(sel(z0, z1)) 212.65/72.39
sel(z0, mark(z1)) → mark(sel(z0, z1)) 212.65/72.39
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 212.65/72.39
proper(f(z0)) → f(proper(z0)) 212.65/72.39
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 212.65/72.39
proper(g(z0)) → g(proper(z0)) 212.65/72.39
proper(0) → ok(0) 212.65/72.39
proper(s(z0)) → s(proper(z0)) 212.65/72.39
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 212.65/72.39
top(mark(z0)) → top(proper(z0)) 212.65/72.39
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.65/72.39
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.65/72.39
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.65/72.39
F(mark(z0)) → c11(F(z0)) 212.65/72.39
F(ok(z0)) → c12(F(z0)) 212.65/72.39
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.65/72.39
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.65/72.39
G(mark(z0)) → c15(G(z0)) 212.65/72.39
G(ok(z0)) → c16(G(z0)) 212.65/72.39
S(mark(z0)) → c17(S(z0)) 212.65/72.39
S(ok(z0)) → c18(S(z0)) 212.65/72.39
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.65/72.39
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.65/72.39
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.65/72.39
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0)) 212.65/72.39
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.65/72.39
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.65/72.39
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.65/72.39
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.65/72.39
ACTIVE(f(z0)) → c(G(z0)) 212.65/72.39
ACTIVE(g(0)) → c1 212.65/72.39
ACTIVE(g(s(z0))) → c2(G(z0)) 212.65/72.39
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.39
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.39
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.39
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.39
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.65/72.39
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.65/72.39
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.65/72.39
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.39
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.39
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.65/72.39
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.65/72.39
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.65/72.39
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.39
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.39
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.39
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.39
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.65/72.39
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.65/72.39
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.65/72.39
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.65/72.39
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.65/72.39
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.65/72.39
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.65/72.39
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.65/72.39
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.65/72.39
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.65/72.39
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.65/72.39
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.65/72.39
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.65/72.39
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.65/72.39
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.65/72.39
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.65/72.39
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.97/72.40
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.97/72.40
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0))) 212.97/72.40
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.40
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0))) 212.97/72.40
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0))) 212.97/72.40
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.40
PROPER(f(0)) → c22(F(ok(0))) 212.97/72.40
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.40
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.40
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.40
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.40
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.40
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.40
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.40
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.40
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.40
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.40
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0)) 212.97/72.40
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)) 212.97/72.40
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.97/72.40
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.97/72.40
F(mark(z0)) → c11(F(z0)) 212.97/72.40
F(ok(z0)) → c12(F(z0)) 212.97/72.40
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.40
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.40
G(mark(z0)) → c15(G(z0)) 212.97/72.40
G(ok(z0)) → c16(G(z0)) 212.97/72.40
S(mark(z0)) → c17(S(z0)) 212.97/72.40
S(ok(z0)) → c18(S(z0)) 212.97/72.40
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.40
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.40
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.97/72.40
PROPER(g(z0)) → c24(G(proper(z0)), PROPER(z0)) 212.97/72.40
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.97/72.40
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.97/72.40
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.97/72.40
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.97/72.40
ACTIVE(f(z0)) → c(G(z0)) 212.97/72.40
ACTIVE(g(0)) → c1 212.97/72.40
ACTIVE(g(s(z0))) → c2(G(z0)) 212.97/72.40
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.40
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.40
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.40
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.40
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.97/72.40
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.97/72.40
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.97/72.40
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.40
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.40
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.97/72.40
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.97/72.40
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.97/72.40
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.40
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.40
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.40
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.40
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.97/72.40
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.97/72.40
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.97/72.40
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.97/72.40
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.97/72.40
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.97/72.40
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.97/72.40
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.97/72.40
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.40
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.40
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.40
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.40
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.97/72.40
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.97/72.40
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.97/72.40
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.97/72.40
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0))) 212.97/72.40
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.40
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0))) 212.97/72.40
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0))) 212.97/72.40
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.40
PROPER(f(0)) → c22(F(ok(0))) 212.97/72.40
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.40
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.40
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.40
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.40
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.40
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.40
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.40
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.40
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.40
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.40
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0)) 212.97/72.40
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1))

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

212.97/72.40

(39) 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))) 212.97/72.40
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.40
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0))) 212.97/72.40
PROPER(g(0)) → c24(G(ok(0)), PROPER(0)) 212.97/72.40
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0))) 212.97/72.40
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))

212.97/72.40

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0)))) 212.97/72.40
active(g(0)) → mark(s(0)) 212.97/72.40
active(g(s(z0))) → mark(s(s(g(z0)))) 212.97/72.40
active(sel(0, cons(z0, z1))) → mark(z0) 212.97/72.40
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 212.97/72.40
active(f(z0)) → f(active(z0)) 212.97/72.40
active(cons(z0, z1)) → cons(active(z0), z1) 212.97/72.40
active(g(z0)) → g(active(z0)) 212.97/72.40
active(s(z0)) → s(active(z0)) 212.97/72.40
active(sel(z0, z1)) → sel(active(z0), z1) 212.97/72.40
active(sel(z0, z1)) → sel(z0, active(z1)) 212.97/72.40
f(mark(z0)) → mark(f(z0)) 212.97/72.40
f(ok(z0)) → ok(f(z0)) 212.97/72.40
cons(mark(z0), z1) → mark(cons(z0, z1)) 212.97/72.40
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 212.97/72.40
g(mark(z0)) → mark(g(z0)) 212.97/72.40
g(ok(z0)) → ok(g(z0)) 212.97/72.40
s(mark(z0)) → mark(s(z0)) 212.97/72.40
s(ok(z0)) → ok(s(z0)) 212.97/72.40
sel(mark(z0), z1) → mark(sel(z0, z1)) 212.97/72.40
sel(z0, mark(z1)) → mark(sel(z0, z1)) 212.97/72.40
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 212.97/72.40
proper(f(z0)) → f(proper(z0)) 212.97/72.40
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 212.97/72.40
proper(g(z0)) → g(proper(z0)) 212.97/72.40
proper(0) → ok(0) 212.97/72.40
proper(s(z0)) → s(proper(z0)) 212.97/72.40
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 212.97/72.40
top(mark(z0)) → top(proper(z0)) 212.97/72.40
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.97/72.40
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.97/72.40
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.97/72.40
F(mark(z0)) → c11(F(z0)) 212.97/72.40
F(ok(z0)) → c12(F(z0)) 212.97/72.40
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.40
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.40
G(mark(z0)) → c15(G(z0)) 212.97/72.40
G(ok(z0)) → c16(G(z0)) 212.97/72.40
S(mark(z0)) → c17(S(z0)) 212.97/72.40
S(ok(z0)) → c18(S(z0)) 212.97/72.40
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.40
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.40
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.97/72.40
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.97/72.40
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.97/72.40
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.97/72.40
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.97/72.40
ACTIVE(f(z0)) → c(G(z0)) 212.97/72.40
ACTIVE(g(0)) → c1 212.97/72.40
ACTIVE(g(s(z0))) → c2(G(z0)) 212.97/72.40
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.40
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.40
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.40
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.40
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.97/72.40
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.97/72.40
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.97/72.40
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.40
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.40
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.97/72.40
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.97/72.40
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.97/72.40
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.40
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.40
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.40
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.40
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.97/72.40
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.97/72.40
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.97/72.40
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.97/72.40
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.97/72.40
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.97/72.40
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.97/72.40
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.97/72.40
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.40
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.40
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.40
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.40
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.97/72.40
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.97/72.40
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.97/72.40
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.97/72.40
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0))) 212.97/72.40
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.40
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0))) 212.97/72.40
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0))) 212.97/72.40
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.40
PROPER(f(0)) → c22(F(ok(0))) 212.97/72.40
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.40
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.40
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.40
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.40
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.40
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.40
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.40
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.40
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.40
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.40
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0)) 212.97/72.40
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1)) 212.97/72.40
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0))) 212.97/72.40
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.40
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0))) 212.97/72.40
PROPER(g(0)) → c24(G(ok(0)), PROPER(0)) 212.97/72.40
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0))) 212.97/72.40
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)) 212.97/72.40
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.97/72.40
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.97/72.40
F(mark(z0)) → c11(F(z0)) 212.97/72.40
F(ok(z0)) → c12(F(z0)) 212.97/72.40
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.40
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.40
G(mark(z0)) → c15(G(z0)) 212.97/72.40
G(ok(z0)) → c16(G(z0)) 212.97/72.40
S(mark(z0)) → c17(S(z0)) 212.97/72.40
S(ok(z0)) → c18(S(z0)) 212.97/72.40
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.40
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.40
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.97/72.40
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.97/72.40
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.97/72.40
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.97/72.40
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.97/72.40
ACTIVE(f(z0)) → c(G(z0)) 212.97/72.40
ACTIVE(g(0)) → c1 212.97/72.40
ACTIVE(g(s(z0))) → c2(G(z0)) 212.97/72.40
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.40
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.40
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.40
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.40
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.97/72.40
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.97/72.40
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.97/72.40
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.40
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.40
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.97/72.40
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.97/72.40
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.97/72.40
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.40
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.40
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.40
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.40
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.97/72.40
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.97/72.40
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.97/72.40
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.97/72.40
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.97/72.40
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.97/72.40
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.97/72.40
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.97/72.40
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.40
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.40
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.40
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.40
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.97/72.40
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.97/72.40
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.97/72.40
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.97/72.40
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0))) 212.97/72.40
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.40
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0))) 212.97/72.40
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0))) 212.97/72.40
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.40
PROPER(f(0)) → c22(F(ok(0))) 212.97/72.40
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.40
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.40
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.40
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.40
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.40
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.40
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.40
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.40
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.40
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.40
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0)) 212.97/72.40
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1)) 212.97/72.40
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0))) 212.97/72.40
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.40
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0))) 212.97/72.40
PROPER(g(0)) → c24(G(ok(0)), PROPER(0)) 212.97/72.40
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0))) 212.97/72.40
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))

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

212.97/72.40

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

Removed 1 trailing tuple parts

212.97/72.40

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0)))) 212.97/72.40
active(g(0)) → mark(s(0)) 212.97/72.40
active(g(s(z0))) → mark(s(s(g(z0)))) 212.97/72.40
active(sel(0, cons(z0, z1))) → mark(z0) 212.97/72.40
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 212.97/72.40
active(f(z0)) → f(active(z0)) 212.97/72.40
active(cons(z0, z1)) → cons(active(z0), z1) 212.97/72.40
active(g(z0)) → g(active(z0)) 212.97/72.40
active(s(z0)) → s(active(z0)) 212.97/72.40
active(sel(z0, z1)) → sel(active(z0), z1) 212.97/72.40
active(sel(z0, z1)) → sel(z0, active(z1)) 212.97/72.40
f(mark(z0)) → mark(f(z0)) 212.97/72.40
f(ok(z0)) → ok(f(z0)) 212.97/72.40
cons(mark(z0), z1) → mark(cons(z0, z1)) 212.97/72.40
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 212.97/72.40
g(mark(z0)) → mark(g(z0)) 212.97/72.40
g(ok(z0)) → ok(g(z0)) 212.97/72.40
s(mark(z0)) → mark(s(z0)) 212.97/72.40
s(ok(z0)) → ok(s(z0)) 212.97/72.40
sel(mark(z0), z1) → mark(sel(z0, z1)) 212.97/72.40
sel(z0, mark(z1)) → mark(sel(z0, z1)) 212.97/72.40
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 212.97/72.40
proper(f(z0)) → f(proper(z0)) 212.97/72.40
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 212.97/72.40
proper(g(z0)) → g(proper(z0)) 212.97/72.40
proper(0) → ok(0) 212.97/72.40
proper(s(z0)) → s(proper(z0)) 212.97/72.40
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 212.97/72.40
top(mark(z0)) → top(proper(z0)) 212.97/72.40
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.97/72.40
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.97/72.40
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.97/72.40
F(mark(z0)) → c11(F(z0)) 212.97/72.40
F(ok(z0)) → c12(F(z0)) 212.97/72.40
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.40
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.40
G(mark(z0)) → c15(G(z0)) 212.97/72.40
G(ok(z0)) → c16(G(z0)) 212.97/72.40
S(mark(z0)) → c17(S(z0)) 212.97/72.40
S(ok(z0)) → c18(S(z0)) 212.97/72.40
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.40
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.40
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.97/72.40
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.97/72.40
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.97/72.40
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.97/72.40
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.97/72.40
ACTIVE(f(z0)) → c(G(z0)) 212.97/72.40
ACTIVE(g(0)) → c1 212.97/72.40
ACTIVE(g(s(z0))) → c2(G(z0)) 212.97/72.40
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.40
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.40
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.40
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.40
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.97/72.40
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.97/72.40
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.97/72.40
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.40
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.40
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.97/72.40
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.97/72.40
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.97/72.40
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.40
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.40
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.40
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.40
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.97/72.40
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.97/72.40
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.97/72.40
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.97/72.40
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.97/72.40
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.97/72.40
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.97/72.40
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.97/72.40
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.40
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.40
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.40
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.40
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.97/72.40
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.97/72.40
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.97/72.40
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.97/72.40
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0))) 212.97/72.40
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.40
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0))) 212.97/72.40
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0))) 212.97/72.40
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.40
PROPER(f(0)) → c22(F(ok(0))) 212.97/72.40
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.40
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.40
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.40
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.40
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.40
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.40
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.40
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.40
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.40
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.40
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0)) 212.97/72.40
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1)) 212.97/72.40
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0))) 212.97/72.40
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.40
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0))) 212.97/72.40
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0))) 212.97/72.40
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.40
PROPER(g(0)) → c24(G(ok(0)))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.97/72.40
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.97/72.40
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.97/72.40
F(mark(z0)) → c11(F(z0)) 212.97/72.40
F(ok(z0)) → c12(F(z0)) 212.97/72.40
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.40
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.40
G(mark(z0)) → c15(G(z0)) 212.97/72.40
G(ok(z0)) → c16(G(z0)) 212.97/72.40
S(mark(z0)) → c17(S(z0)) 212.97/72.40
S(ok(z0)) → c18(S(z0)) 212.97/72.40
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.40
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.40
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.97/72.40
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.97/72.40
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.97/72.40
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.97/72.40
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.97/72.40
ACTIVE(f(z0)) → c(G(z0)) 212.97/72.40
ACTIVE(g(0)) → c1 212.97/72.40
ACTIVE(g(s(z0))) → c2(G(z0)) 212.97/72.40
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.40
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.40
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.40
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.40
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.97/72.40
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.97/72.40
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.97/72.40
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.40
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.40
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.97/72.40
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.97/72.40
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.97/72.40
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.40
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.40
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.40
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.40
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.97/72.40
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.97/72.40
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.97/72.40
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.97/72.40
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.97/72.40
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.97/72.40
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.97/72.40
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.97/72.40
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.40
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.40
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.40
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.40
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.97/72.40
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.97/72.40
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.97/72.40
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.97/72.40
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0))) 212.97/72.40
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.40
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0))) 212.97/72.40
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0))) 212.97/72.40
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.40
PROPER(f(0)) → c22(F(ok(0))) 212.97/72.40
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.40
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.40
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.40
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.40
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.40
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.40
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.40
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.40
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.40
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.40
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0)) 212.97/72.40
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1)) 212.97/72.40
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0))) 212.97/72.40
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.40
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0))) 212.97/72.40
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0))) 212.97/72.40
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.40
PROPER(g(0)) → c24(G(ok(0)))

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

212.97/72.40

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

Removed 5 trailing nodes:

ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.97/72.40
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.97/72.40
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.97/72.40
ACTIVE(g(0)) → c1 212.97/72.40
ACTIVE(f(g(0))) → c3(ACTIVE(g(0)))

212.97/72.40

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0)))) 212.97/72.40
active(g(0)) → mark(s(0)) 212.97/72.40
active(g(s(z0))) → mark(s(s(g(z0)))) 212.97/72.40
active(sel(0, cons(z0, z1))) → mark(z0) 212.97/72.40
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 212.97/72.40
active(f(z0)) → f(active(z0)) 212.97/72.40
active(cons(z0, z1)) → cons(active(z0), z1) 212.97/72.40
active(g(z0)) → g(active(z0)) 212.97/72.40
active(s(z0)) → s(active(z0)) 212.97/72.40
active(sel(z0, z1)) → sel(active(z0), z1) 212.97/72.40
active(sel(z0, z1)) → sel(z0, active(z1)) 212.97/72.40
f(mark(z0)) → mark(f(z0)) 212.97/72.40
f(ok(z0)) → ok(f(z0)) 212.97/72.40
cons(mark(z0), z1) → mark(cons(z0, z1)) 212.97/72.40
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 212.97/72.40
g(mark(z0)) → mark(g(z0)) 212.97/72.40
g(ok(z0)) → ok(g(z0)) 212.97/72.40
s(mark(z0)) → mark(s(z0)) 212.97/72.40
s(ok(z0)) → ok(s(z0)) 212.97/72.40
sel(mark(z0), z1) → mark(sel(z0, z1)) 212.97/72.40
sel(z0, mark(z1)) → mark(sel(z0, z1)) 212.97/72.40
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 212.97/72.40
proper(f(z0)) → f(proper(z0)) 212.97/72.40
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 212.97/72.40
proper(g(z0)) → g(proper(z0)) 212.97/72.40
proper(0) → ok(0) 212.97/72.40
proper(s(z0)) → s(proper(z0)) 212.97/72.40
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 212.97/72.40
top(mark(z0)) → top(proper(z0)) 212.97/72.40
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.97/72.40
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.97/72.40
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.97/72.40
F(mark(z0)) → c11(F(z0)) 212.97/72.40
F(ok(z0)) → c12(F(z0)) 212.97/72.40
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.40
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.40
G(mark(z0)) → c15(G(z0)) 212.97/72.40
G(ok(z0)) → c16(G(z0)) 212.97/72.40
S(mark(z0)) → c17(S(z0)) 212.97/72.40
S(ok(z0)) → c18(S(z0)) 212.97/72.40
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.40
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.40
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.97/72.40
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.97/72.40
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.97/72.40
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.97/72.40
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.97/72.40
ACTIVE(f(z0)) → c(G(z0)) 212.97/72.40
ACTIVE(g(0)) → c1 212.97/72.40
ACTIVE(g(s(z0))) → c2(G(z0)) 212.97/72.40
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.40
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.40
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.40
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.40
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.97/72.40
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.97/72.40
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.97/72.40
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.40
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.40
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.97/72.40
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.97/72.40
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.97/72.40
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.40
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.40
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.40
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.40
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.97/72.40
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.97/72.40
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.97/72.40
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.97/72.40
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.97/72.40
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.97/72.40
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.97/72.40
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.97/72.40
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.40
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.40
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.40
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.40
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.97/72.40
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.97/72.40
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.97/72.40
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.97/72.40
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0))) 212.97/72.40
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.40
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0))) 212.97/72.40
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0))) 212.97/72.40
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.40
PROPER(f(0)) → c22(F(ok(0))) 212.97/72.40
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.40
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.40
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.40
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.40
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.40
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.40
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.40
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.40
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.40
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.40
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0)) 212.97/72.40
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1)) 212.97/72.40
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0))) 212.97/72.40
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.40
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0))) 212.97/72.40
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0))) 212.97/72.40
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.40
PROPER(g(0)) → c24(G(ok(0)))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.97/72.40
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.97/72.40
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.97/72.40
F(mark(z0)) → c11(F(z0)) 212.97/72.40
F(ok(z0)) → c12(F(z0)) 212.97/72.40
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.40
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.40
G(mark(z0)) → c15(G(z0)) 212.97/72.40
G(ok(z0)) → c16(G(z0)) 212.97/72.40
S(mark(z0)) → c17(S(z0)) 212.97/72.40
S(ok(z0)) → c18(S(z0)) 212.97/72.40
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.40
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.40
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.97/72.40
PROPER(s(z0)) → c26(S(proper(z0)), PROPER(z0)) 212.97/72.40
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.97/72.40
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.97/72.40
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.97/72.40
ACTIVE(f(z0)) → c(G(z0)) 212.97/72.40
ACTIVE(g(0)) → c1 212.97/72.40
ACTIVE(g(s(z0))) → c2(G(z0)) 212.97/72.40
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.40
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.40
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.40
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.40
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.97/72.40
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.97/72.40
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.97/72.40
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.40
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.40
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.97/72.40
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.97/72.40
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.97/72.40
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.40
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.40
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.40
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.40
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.97/72.40
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.97/72.40
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.97/72.40
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.97/72.40
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.97/72.40
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.97/72.40
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.97/72.40
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.97/72.40
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.40
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.40
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.40
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.40
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.97/72.40
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.97/72.40
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.97/72.40
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.97/72.40
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0))) 212.97/72.40
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.40
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0))) 212.97/72.40
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0))) 212.97/72.40
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.40
PROPER(f(0)) → c22(F(ok(0))) 212.97/72.40
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.40
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.40
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.40
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.40
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.40
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.40
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.40
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.40
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.40
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.40
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0)) 212.97/72.40
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1)) 212.97/72.40
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0))) 212.97/72.40
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.40
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0))) 212.97/72.40
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0))) 212.97/72.40
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.40
PROPER(g(0)) → c24(G(ok(0)))

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

212.97/72.40

(45) 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))) 212.97/72.40
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.40
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0))) 212.97/72.40
PROPER(s(0)) → c26(S(ok(0)), PROPER(0)) 212.97/72.40
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 212.97/72.40
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))

212.97/72.40

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0)))) 212.97/72.40
active(g(0)) → mark(s(0)) 212.97/72.40
active(g(s(z0))) → mark(s(s(g(z0)))) 212.97/72.40
active(sel(0, cons(z0, z1))) → mark(z0) 212.97/72.40
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 212.97/72.40
active(f(z0)) → f(active(z0)) 212.97/72.40
active(cons(z0, z1)) → cons(active(z0), z1) 212.97/72.40
active(g(z0)) → g(active(z0)) 212.97/72.40
active(s(z0)) → s(active(z0)) 212.97/72.40
active(sel(z0, z1)) → sel(active(z0), z1) 212.97/72.40
active(sel(z0, z1)) → sel(z0, active(z1)) 212.97/72.40
f(mark(z0)) → mark(f(z0)) 212.97/72.40
f(ok(z0)) → ok(f(z0)) 212.97/72.40
cons(mark(z0), z1) → mark(cons(z0, z1)) 212.97/72.40
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 212.97/72.40
g(mark(z0)) → mark(g(z0)) 212.97/72.40
g(ok(z0)) → ok(g(z0)) 212.97/72.40
s(mark(z0)) → mark(s(z0)) 212.97/72.40
s(ok(z0)) → ok(s(z0)) 212.97/72.40
sel(mark(z0), z1) → mark(sel(z0, z1)) 212.97/72.40
sel(z0, mark(z1)) → mark(sel(z0, z1)) 212.97/72.40
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 212.97/72.40
proper(f(z0)) → f(proper(z0)) 212.97/72.40
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 212.97/72.40
proper(g(z0)) → g(proper(z0)) 212.97/72.40
proper(0) → ok(0) 212.97/72.40
proper(s(z0)) → s(proper(z0)) 212.97/72.40
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 212.97/72.40
top(mark(z0)) → top(proper(z0)) 212.97/72.40
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.97/72.40
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.97/72.40
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.97/72.40
F(mark(z0)) → c11(F(z0)) 212.97/72.40
F(ok(z0)) → c12(F(z0)) 212.97/72.40
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.40
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.40
G(mark(z0)) → c15(G(z0)) 212.97/72.40
G(ok(z0)) → c16(G(z0)) 212.97/72.40
S(mark(z0)) → c17(S(z0)) 212.97/72.40
S(ok(z0)) → c18(S(z0)) 212.97/72.40
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.40
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.40
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.97/72.40
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.97/72.40
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.97/72.40
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.97/72.40
ACTIVE(f(z0)) → c(G(z0)) 212.97/72.40
ACTIVE(g(0)) → c1 212.97/72.40
ACTIVE(g(s(z0))) → c2(G(z0)) 212.97/72.40
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.40
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.40
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.40
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.40
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.97/72.40
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.97/72.40
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.97/72.40
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.40
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.40
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.97/72.40
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.97/72.40
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.97/72.40
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.40
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.40
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.40
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.40
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.97/72.40
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.97/72.40
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.97/72.40
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.97/72.40
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.97/72.40
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.97/72.40
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.97/72.40
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.97/72.40
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.40
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.40
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.40
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.97/72.40
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.40
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.97/72.40
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.97/72.40
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.40
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.97/72.40
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.97/72.40
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0))) 212.97/72.40
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0))) 212.97/72.41
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0))) 212.97/72.41
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(f(0)) → c22(F(ok(0))) 212.97/72.41
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.41
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.41
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.41
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.41
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.41
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.41
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.41
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.41
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0)) 212.97/72.41
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1)) 212.97/72.41
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0))) 212.97/72.41
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0))) 212.97/72.41
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0))) 212.97/72.41
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(g(0)) → c24(G(ok(0))) 212.97/72.41
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0))) 212.97/72.41
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0))) 212.97/72.41
PROPER(s(0)) → c26(S(ok(0)), PROPER(0)) 212.97/72.41
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 212.97/72.41
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)) 212.97/72.41
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.97/72.41
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.97/72.41
F(mark(z0)) → c11(F(z0)) 212.97/72.41
F(ok(z0)) → c12(F(z0)) 212.97/72.41
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.41
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.41
G(mark(z0)) → c15(G(z0)) 212.97/72.41
G(ok(z0)) → c16(G(z0)) 212.97/72.41
S(mark(z0)) → c17(S(z0)) 212.97/72.41
S(ok(z0)) → c18(S(z0)) 212.97/72.41
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.41
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.41
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.97/72.41
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.97/72.41
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.97/72.41
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.97/72.41
ACTIVE(f(z0)) → c(G(z0)) 212.97/72.41
ACTIVE(g(0)) → c1 212.97/72.41
ACTIVE(g(s(z0))) → c2(G(z0)) 212.97/72.41
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.97/72.41
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.97/72.41
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.97/72.41
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.97/72.41
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.97/72.41
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.97/72.41
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.97/72.41
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.97/72.41
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0))) 212.97/72.41
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0))) 212.97/72.41
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0))) 212.97/72.41
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(f(0)) → c22(F(ok(0))) 212.97/72.41
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.41
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.41
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.41
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.41
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.41
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.41
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.41
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.41
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0)) 212.97/72.41
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1)) 212.97/72.41
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0))) 212.97/72.41
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0))) 212.97/72.41
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0))) 212.97/72.41
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(g(0)) → c24(G(ok(0))) 212.97/72.41
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0))) 212.97/72.41
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0))) 212.97/72.41
PROPER(s(0)) → c26(S(ok(0)), PROPER(0)) 212.97/72.41
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 212.97/72.41
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))

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

212.97/72.41

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

Removed 1 trailing tuple parts

212.97/72.41

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0)))) 212.97/72.41
active(g(0)) → mark(s(0)) 212.97/72.41
active(g(s(z0))) → mark(s(s(g(z0)))) 212.97/72.41
active(sel(0, cons(z0, z1))) → mark(z0) 212.97/72.41
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 212.97/72.41
active(f(z0)) → f(active(z0)) 212.97/72.41
active(cons(z0, z1)) → cons(active(z0), z1) 212.97/72.41
active(g(z0)) → g(active(z0)) 212.97/72.41
active(s(z0)) → s(active(z0)) 212.97/72.41
active(sel(z0, z1)) → sel(active(z0), z1) 212.97/72.41
active(sel(z0, z1)) → sel(z0, active(z1)) 212.97/72.41
f(mark(z0)) → mark(f(z0)) 212.97/72.41
f(ok(z0)) → ok(f(z0)) 212.97/72.41
cons(mark(z0), z1) → mark(cons(z0, z1)) 212.97/72.41
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 212.97/72.41
g(mark(z0)) → mark(g(z0)) 212.97/72.41
g(ok(z0)) → ok(g(z0)) 212.97/72.41
s(mark(z0)) → mark(s(z0)) 212.97/72.41
s(ok(z0)) → ok(s(z0)) 212.97/72.41
sel(mark(z0), z1) → mark(sel(z0, z1)) 212.97/72.41
sel(z0, mark(z1)) → mark(sel(z0, z1)) 212.97/72.41
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 212.97/72.41
proper(f(z0)) → f(proper(z0)) 212.97/72.41
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 212.97/72.41
proper(g(z0)) → g(proper(z0)) 212.97/72.41
proper(0) → ok(0) 212.97/72.41
proper(s(z0)) → s(proper(z0)) 212.97/72.41
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 212.97/72.41
top(mark(z0)) → top(proper(z0)) 212.97/72.41
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.97/72.41
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.97/72.41
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.97/72.41
F(mark(z0)) → c11(F(z0)) 212.97/72.41
F(ok(z0)) → c12(F(z0)) 212.97/72.41
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.41
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.41
G(mark(z0)) → c15(G(z0)) 212.97/72.41
G(ok(z0)) → c16(G(z0)) 212.97/72.41
S(mark(z0)) → c17(S(z0)) 212.97/72.41
S(ok(z0)) → c18(S(z0)) 212.97/72.41
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.41
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.41
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.97/72.41
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.97/72.41
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.97/72.41
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.97/72.41
ACTIVE(f(z0)) → c(G(z0)) 212.97/72.41
ACTIVE(g(0)) → c1 212.97/72.41
ACTIVE(g(s(z0))) → c2(G(z0)) 212.97/72.41
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.97/72.41
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.97/72.41
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.97/72.41
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.97/72.41
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.97/72.41
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.97/72.41
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.97/72.41
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.97/72.41
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0))) 212.97/72.41
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0))) 212.97/72.41
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0))) 212.97/72.41
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(f(0)) → c22(F(ok(0))) 212.97/72.41
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.41
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.41
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.41
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.41
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.41
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.41
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.41
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.41
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0)) 212.97/72.41
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1)) 212.97/72.41
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0))) 212.97/72.41
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0))) 212.97/72.41
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0))) 212.97/72.41
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(g(0)) → c24(G(ok(0))) 212.97/72.41
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0))) 212.97/72.41
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0))) 212.97/72.41
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 212.97/72.41
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(s(0)) → c26(S(ok(0)))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.97/72.41
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.97/72.41
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.97/72.41
F(mark(z0)) → c11(F(z0)) 212.97/72.41
F(ok(z0)) → c12(F(z0)) 212.97/72.41
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.41
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.41
G(mark(z0)) → c15(G(z0)) 212.97/72.41
G(ok(z0)) → c16(G(z0)) 212.97/72.41
S(mark(z0)) → c17(S(z0)) 212.97/72.41
S(ok(z0)) → c18(S(z0)) 212.97/72.41
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.41
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.41
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.97/72.41
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.97/72.41
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.97/72.41
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.97/72.41
ACTIVE(f(z0)) → c(G(z0)) 212.97/72.41
ACTIVE(g(0)) → c1 212.97/72.41
ACTIVE(g(s(z0))) → c2(G(z0)) 212.97/72.41
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.97/72.41
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.97/72.41
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.97/72.41
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.97/72.41
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.97/72.41
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.97/72.41
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.97/72.41
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.97/72.41
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0))) 212.97/72.41
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0))) 212.97/72.41
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0))) 212.97/72.41
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(f(0)) → c22(F(ok(0))) 212.97/72.41
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.41
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.41
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.41
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.41
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.41
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.41
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.41
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.41
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0)) 212.97/72.41
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1)) 212.97/72.41
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0))) 212.97/72.41
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0))) 212.97/72.41
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0))) 212.97/72.41
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(g(0)) → c24(G(ok(0))) 212.97/72.41
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0))) 212.97/72.41
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0))) 212.97/72.41
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 212.97/72.41
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(s(0)) → c26(S(ok(0)))

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

212.97/72.41

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

Removed 5 trailing nodes:

ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.97/72.41
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.97/72.41
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.97/72.41
ACTIVE(g(0)) → c1 212.97/72.41
ACTIVE(f(g(0))) → c3(ACTIVE(g(0)))

212.97/72.41

(50) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0)))) 212.97/72.41
active(g(0)) → mark(s(0)) 212.97/72.41
active(g(s(z0))) → mark(s(s(g(z0)))) 212.97/72.41
active(sel(0, cons(z0, z1))) → mark(z0) 212.97/72.41
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 212.97/72.41
active(f(z0)) → f(active(z0)) 212.97/72.41
active(cons(z0, z1)) → cons(active(z0), z1) 212.97/72.41
active(g(z0)) → g(active(z0)) 212.97/72.41
active(s(z0)) → s(active(z0)) 212.97/72.41
active(sel(z0, z1)) → sel(active(z0), z1) 212.97/72.41
active(sel(z0, z1)) → sel(z0, active(z1)) 212.97/72.41
f(mark(z0)) → mark(f(z0)) 212.97/72.41
f(ok(z0)) → ok(f(z0)) 212.97/72.41
cons(mark(z0), z1) → mark(cons(z0, z1)) 212.97/72.41
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 212.97/72.41
g(mark(z0)) → mark(g(z0)) 212.97/72.41
g(ok(z0)) → ok(g(z0)) 212.97/72.41
s(mark(z0)) → mark(s(z0)) 212.97/72.41
s(ok(z0)) → ok(s(z0)) 212.97/72.41
sel(mark(z0), z1) → mark(sel(z0, z1)) 212.97/72.41
sel(z0, mark(z1)) → mark(sel(z0, z1)) 212.97/72.41
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 212.97/72.41
proper(f(z0)) → f(proper(z0)) 212.97/72.41
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 212.97/72.41
proper(g(z0)) → g(proper(z0)) 212.97/72.41
proper(0) → ok(0) 212.97/72.41
proper(s(z0)) → s(proper(z0)) 212.97/72.41
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 212.97/72.41
top(mark(z0)) → top(proper(z0)) 212.97/72.41
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.97/72.41
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.97/72.41
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.97/72.41
F(mark(z0)) → c11(F(z0)) 212.97/72.41
F(ok(z0)) → c12(F(z0)) 212.97/72.41
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.41
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.41
G(mark(z0)) → c15(G(z0)) 212.97/72.41
G(ok(z0)) → c16(G(z0)) 212.97/72.41
S(mark(z0)) → c17(S(z0)) 212.97/72.41
S(ok(z0)) → c18(S(z0)) 212.97/72.41
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.41
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.41
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.97/72.41
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.97/72.41
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.97/72.41
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.97/72.41
ACTIVE(f(z0)) → c(G(z0)) 212.97/72.41
ACTIVE(g(0)) → c1 212.97/72.41
ACTIVE(g(s(z0))) → c2(G(z0)) 212.97/72.41
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.97/72.41
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.97/72.41
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.97/72.41
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.97/72.41
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.97/72.41
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.97/72.41
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.97/72.41
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.97/72.41
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0))) 212.97/72.41
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0))) 212.97/72.41
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0))) 212.97/72.41
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(f(0)) → c22(F(ok(0))) 212.97/72.41
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.41
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.41
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.41
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.41
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.41
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.41
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.41
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.41
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0)) 212.97/72.41
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1)) 212.97/72.41
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0))) 212.97/72.41
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0))) 212.97/72.41
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0))) 212.97/72.41
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(g(0)) → c24(G(ok(0))) 212.97/72.41
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0))) 212.97/72.41
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0))) 212.97/72.41
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 212.97/72.41
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(s(0)) → c26(S(ok(0)))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.97/72.41
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.97/72.41
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.97/72.41
F(mark(z0)) → c11(F(z0)) 212.97/72.41
F(ok(z0)) → c12(F(z0)) 212.97/72.41
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.41
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.41
G(mark(z0)) → c15(G(z0)) 212.97/72.41
G(ok(z0)) → c16(G(z0)) 212.97/72.41
S(mark(z0)) → c17(S(z0)) 212.97/72.41
S(ok(z0)) → c18(S(z0)) 212.97/72.41
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.41
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.41
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.97/72.41
PROPER(sel(z0, z1)) → c27(SEL(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 212.97/72.41
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.97/72.41
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.97/72.41
ACTIVE(f(z0)) → c(G(z0)) 212.97/72.41
ACTIVE(g(0)) → c1 212.97/72.41
ACTIVE(g(s(z0))) → c2(G(z0)) 212.97/72.41
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.97/72.41
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.97/72.41
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.97/72.41
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.97/72.41
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.97/72.41
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.97/72.41
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.97/72.41
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.97/72.41
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0))) 212.97/72.41
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0))) 212.97/72.41
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0))) 212.97/72.41
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(f(0)) → c22(F(ok(0))) 212.97/72.41
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.41
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.41
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.41
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.41
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.41
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.41
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.41
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.41
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0)) 212.97/72.41
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1)) 212.97/72.41
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0))) 212.97/72.41
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0))) 212.97/72.41
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0))) 212.97/72.41
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(g(0)) → c24(G(ok(0))) 212.97/72.41
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0))) 212.97/72.41
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0))) 212.97/72.41
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 212.97/72.41
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(s(0)) → c26(S(ok(0)))

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

212.97/72.41

(51) 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))) 212.97/72.41
PROPER(sel(x0, cons(z0, z1))) → c27(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(sel(x0, g(z0))) → c27(SEL(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.41
PROPER(sel(x0, 0)) → c27(SEL(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 212.97/72.41
PROPER(sel(x0, s(z0))) → c27(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.41
PROPER(sel(x0, sel(z0, z1))) → c27(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(sel(f(z0), x1)) → c27(SEL(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.41
PROPER(sel(cons(z0, z1), x1)) → c27(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.41
PROPER(sel(g(z0), x1)) → c27(SEL(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.41
PROPER(sel(0, x1)) → c27(SEL(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 212.97/72.41
PROPER(sel(s(z0), x1)) → c27(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.41
PROPER(sel(sel(z0, z1), x1)) → c27(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1))

212.97/72.41

(52) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0)))) 212.97/72.41
active(g(0)) → mark(s(0)) 212.97/72.41
active(g(s(z0))) → mark(s(s(g(z0)))) 212.97/72.41
active(sel(0, cons(z0, z1))) → mark(z0) 212.97/72.41
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 212.97/72.41
active(f(z0)) → f(active(z0)) 212.97/72.41
active(cons(z0, z1)) → cons(active(z0), z1) 212.97/72.41
active(g(z0)) → g(active(z0)) 212.97/72.41
active(s(z0)) → s(active(z0)) 212.97/72.41
active(sel(z0, z1)) → sel(active(z0), z1) 212.97/72.41
active(sel(z0, z1)) → sel(z0, active(z1)) 212.97/72.41
f(mark(z0)) → mark(f(z0)) 212.97/72.41
f(ok(z0)) → ok(f(z0)) 212.97/72.41
cons(mark(z0), z1) → mark(cons(z0, z1)) 212.97/72.41
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 212.97/72.41
g(mark(z0)) → mark(g(z0)) 212.97/72.41
g(ok(z0)) → ok(g(z0)) 212.97/72.41
s(mark(z0)) → mark(s(z0)) 212.97/72.41
s(ok(z0)) → ok(s(z0)) 212.97/72.41
sel(mark(z0), z1) → mark(sel(z0, z1)) 212.97/72.41
sel(z0, mark(z1)) → mark(sel(z0, z1)) 212.97/72.41
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 212.97/72.41
proper(f(z0)) → f(proper(z0)) 212.97/72.41
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 212.97/72.41
proper(g(z0)) → g(proper(z0)) 212.97/72.41
proper(0) → ok(0) 212.97/72.41
proper(s(z0)) → s(proper(z0)) 212.97/72.41
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 212.97/72.41
top(mark(z0)) → top(proper(z0)) 212.97/72.41
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.97/72.41
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.97/72.41
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.97/72.41
F(mark(z0)) → c11(F(z0)) 212.97/72.41
F(ok(z0)) → c12(F(z0)) 212.97/72.41
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.41
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.41
G(mark(z0)) → c15(G(z0)) 212.97/72.41
G(ok(z0)) → c16(G(z0)) 212.97/72.41
S(mark(z0)) → c17(S(z0)) 212.97/72.41
S(ok(z0)) → c18(S(z0)) 212.97/72.41
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.41
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.41
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.97/72.41
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.97/72.41
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.97/72.41
ACTIVE(f(z0)) → c(G(z0)) 212.97/72.41
ACTIVE(g(0)) → c1 212.97/72.41
ACTIVE(g(s(z0))) → c2(G(z0)) 212.97/72.41
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.97/72.41
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.97/72.41
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.97/72.41
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.97/72.41
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.97/72.41
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.97/72.41
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.97/72.41
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.97/72.41
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0))) 212.97/72.41
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0))) 212.97/72.41
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0))) 212.97/72.41
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(f(0)) → c22(F(ok(0))) 212.97/72.41
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.41
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.41
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.41
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.41
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.41
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.41
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.41
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.41
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0)) 212.97/72.41
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1)) 212.97/72.41
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0))) 212.97/72.41
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0))) 212.97/72.41
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0))) 212.97/72.41
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(g(0)) → c24(G(ok(0))) 212.97/72.41
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0))) 212.97/72.41
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0))) 212.97/72.41
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 212.97/72.41
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(s(0)) → c26(S(ok(0))) 212.97/72.41
PROPER(sel(x0, f(z0))) → c27(SEL(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.41
PROPER(sel(x0, cons(z0, z1))) → c27(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(sel(x0, g(z0))) → c27(SEL(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.41
PROPER(sel(x0, 0)) → c27(SEL(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 212.97/72.41
PROPER(sel(x0, s(z0))) → c27(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.41
PROPER(sel(x0, sel(z0, z1))) → c27(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(sel(f(z0), x1)) → c27(SEL(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.41
PROPER(sel(cons(z0, z1), x1)) → c27(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.41
PROPER(sel(g(z0), x1)) → c27(SEL(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.41
PROPER(sel(0, x1)) → c27(SEL(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 212.97/72.41
PROPER(sel(s(z0), x1)) → c27(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.41
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)) 212.97/72.41
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.97/72.41
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.97/72.41
F(mark(z0)) → c11(F(z0)) 212.97/72.41
F(ok(z0)) → c12(F(z0)) 212.97/72.41
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.41
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.41
G(mark(z0)) → c15(G(z0)) 212.97/72.41
G(ok(z0)) → c16(G(z0)) 212.97/72.41
S(mark(z0)) → c17(S(z0)) 212.97/72.41
S(ok(z0)) → c18(S(z0)) 212.97/72.41
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.41
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.41
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.97/72.41
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.97/72.41
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.97/72.41
ACTIVE(f(z0)) → c(G(z0)) 212.97/72.41
ACTIVE(g(0)) → c1 212.97/72.41
ACTIVE(g(s(z0))) → c2(G(z0)) 212.97/72.41
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.97/72.41
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.97/72.41
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.97/72.41
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.97/72.41
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.97/72.41
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.97/72.41
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.97/72.41
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.97/72.41
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0))) 212.97/72.41
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0))) 212.97/72.41
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0))) 212.97/72.41
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(f(0)) → c22(F(ok(0))) 212.97/72.41
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.41
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.41
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.41
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.41
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.41
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.41
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.41
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.41
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0)) 212.97/72.41
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1)) 212.97/72.41
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0))) 212.97/72.41
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0))) 212.97/72.41
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0))) 212.97/72.41
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(g(0)) → c24(G(ok(0))) 212.97/72.41
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0))) 212.97/72.41
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0))) 212.97/72.41
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 212.97/72.41
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(s(0)) → c26(S(ok(0))) 212.97/72.41
PROPER(sel(x0, f(z0))) → c27(SEL(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.41
PROPER(sel(x0, cons(z0, z1))) → c27(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(sel(x0, g(z0))) → c27(SEL(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.41
PROPER(sel(x0, 0)) → c27(SEL(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 212.97/72.41
PROPER(sel(x0, s(z0))) → c27(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.41
PROPER(sel(x0, sel(z0, z1))) → c27(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(sel(f(z0), x1)) → c27(SEL(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.41
PROPER(sel(cons(z0, z1), x1)) → c27(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.41
PROPER(sel(g(z0), x1)) → c27(SEL(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.41
PROPER(sel(0, x1)) → c27(SEL(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 212.97/72.41
PROPER(sel(s(z0), x1)) → c27(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.41
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, f, cons, g, s, sel, proper, top

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, c1, c2, c5, c6, c7, c3, c8, c25, c22, c22, c23, c23, c24, c24, c26, c26, c27

212.97/72.41

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

Removed 2 trailing tuple parts

212.97/72.41

(54) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0)))) 212.97/72.41
active(g(0)) → mark(s(0)) 212.97/72.41
active(g(s(z0))) → mark(s(s(g(z0)))) 212.97/72.41
active(sel(0, cons(z0, z1))) → mark(z0) 212.97/72.41
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 212.97/72.41
active(f(z0)) → f(active(z0)) 212.97/72.41
active(cons(z0, z1)) → cons(active(z0), z1) 212.97/72.41
active(g(z0)) → g(active(z0)) 212.97/72.41
active(s(z0)) → s(active(z0)) 212.97/72.41
active(sel(z0, z1)) → sel(active(z0), z1) 212.97/72.41
active(sel(z0, z1)) → sel(z0, active(z1)) 212.97/72.41
f(mark(z0)) → mark(f(z0)) 212.97/72.41
f(ok(z0)) → ok(f(z0)) 212.97/72.41
cons(mark(z0), z1) → mark(cons(z0, z1)) 212.97/72.41
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 212.97/72.41
g(mark(z0)) → mark(g(z0)) 212.97/72.41
g(ok(z0)) → ok(g(z0)) 212.97/72.41
s(mark(z0)) → mark(s(z0)) 212.97/72.41
s(ok(z0)) → ok(s(z0)) 212.97/72.41
sel(mark(z0), z1) → mark(sel(z0, z1)) 212.97/72.41
sel(z0, mark(z1)) → mark(sel(z0, z1)) 212.97/72.41
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 212.97/72.41
proper(f(z0)) → f(proper(z0)) 212.97/72.41
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 212.97/72.41
proper(g(z0)) → g(proper(z0)) 212.97/72.41
proper(0) → ok(0) 212.97/72.41
proper(s(z0)) → s(proper(z0)) 212.97/72.41
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 212.97/72.41
top(mark(z0)) → top(proper(z0)) 212.97/72.41
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.97/72.41
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.97/72.41
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.97/72.41
F(mark(z0)) → c11(F(z0)) 212.97/72.41
F(ok(z0)) → c12(F(z0)) 212.97/72.41
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.41
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.41
G(mark(z0)) → c15(G(z0)) 212.97/72.41
G(ok(z0)) → c16(G(z0)) 212.97/72.41
S(mark(z0)) → c17(S(z0)) 212.97/72.41
S(ok(z0)) → c18(S(z0)) 212.97/72.41
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.41
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.41
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.97/72.41
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.97/72.41
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.97/72.41
ACTIVE(f(z0)) → c(G(z0)) 212.97/72.41
ACTIVE(g(0)) → c1 212.97/72.41
ACTIVE(g(s(z0))) → c2(G(z0)) 212.97/72.41
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.97/72.41
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.97/72.41
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.97/72.41
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.97/72.41
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.97/72.41
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.97/72.41
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.97/72.41
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.97/72.41
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0))) 212.97/72.41
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0))) 212.97/72.41
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0))) 212.97/72.41
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(f(0)) → c22(F(ok(0))) 212.97/72.41
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.41
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.41
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.41
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.41
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.41
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.41
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.41
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.41
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0)) 212.97/72.41
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1)) 212.97/72.41
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0))) 212.97/72.41
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0))) 212.97/72.41
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0))) 212.97/72.41
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(g(0)) → c24(G(ok(0))) 212.97/72.41
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0))) 212.97/72.41
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0))) 212.97/72.41
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 212.97/72.41
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(s(0)) → c26(S(ok(0))) 212.97/72.41
PROPER(sel(x0, f(z0))) → c27(SEL(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.41
PROPER(sel(x0, cons(z0, z1))) → c27(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(sel(x0, g(z0))) → c27(SEL(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.41
PROPER(sel(x0, s(z0))) → c27(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.41
PROPER(sel(x0, sel(z0, z1))) → c27(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(sel(f(z0), x1)) → c27(SEL(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.41
PROPER(sel(cons(z0, z1), x1)) → c27(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.41
PROPER(sel(g(z0), x1)) → c27(SEL(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.41
PROPER(sel(s(z0), x1)) → c27(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.41
PROPER(sel(sel(z0, z1), x1)) → c27(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.41
PROPER(sel(x0, 0)) → c27(SEL(proper(x0), ok(0)), PROPER(x0)) 212.97/72.41
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)) 212.97/72.41
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.97/72.41
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.97/72.41
F(mark(z0)) → c11(F(z0)) 212.97/72.41
F(ok(z0)) → c12(F(z0)) 212.97/72.41
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.41
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.41
G(mark(z0)) → c15(G(z0)) 212.97/72.41
G(ok(z0)) → c16(G(z0)) 212.97/72.41
S(mark(z0)) → c17(S(z0)) 212.97/72.41
S(ok(z0)) → c18(S(z0)) 212.97/72.41
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.41
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.41
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.97/72.41
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.97/72.41
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.97/72.41
ACTIVE(f(z0)) → c(G(z0)) 212.97/72.41
ACTIVE(g(0)) → c1 212.97/72.41
ACTIVE(g(s(z0))) → c2(G(z0)) 212.97/72.41
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.97/72.41
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.97/72.41
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.97/72.41
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.97/72.41
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.97/72.41
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.97/72.41
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.97/72.41
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.97/72.41
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0))) 212.97/72.41
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0))) 212.97/72.41
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0))) 212.97/72.41
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(f(0)) → c22(F(ok(0))) 212.97/72.41
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.41
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.41
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.41
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.41
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.41
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.41
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.41
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.41
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0)) 212.97/72.41
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1)) 212.97/72.41
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0))) 212.97/72.41
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0))) 212.97/72.41
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0))) 212.97/72.41
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(g(0)) → c24(G(ok(0))) 212.97/72.41
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0))) 212.97/72.41
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0))) 212.97/72.41
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 212.97/72.41
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(s(0)) → c26(S(ok(0))) 212.97/72.41
PROPER(sel(x0, f(z0))) → c27(SEL(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.41
PROPER(sel(x0, cons(z0, z1))) → c27(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(sel(x0, g(z0))) → c27(SEL(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.41
PROPER(sel(x0, s(z0))) → c27(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.41
PROPER(sel(x0, sel(z0, z1))) → c27(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(sel(f(z0), x1)) → c27(SEL(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.41
PROPER(sel(cons(z0, z1), x1)) → c27(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.41
PROPER(sel(g(z0), x1)) → c27(SEL(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.41
PROPER(sel(s(z0), x1)) → c27(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.41
PROPER(sel(sel(z0, z1), x1)) → c27(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.41
PROPER(sel(x0, 0)) → c27(SEL(proper(x0), ok(0)), PROPER(x0)) 212.97/72.41
PROPER(sel(0, x1)) → c27(SEL(ok(0), proper(x1)), PROPER(x1))

K tuples:none
Defined Rule Symbols:

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

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, c1, c2, c5, c6, c7, c3, c8, c25, c22, c22, c23, c23, c24, c24, c26, c26, c27, c27

212.97/72.41

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

Removed 5 trailing nodes:

ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.97/72.41
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.97/72.41
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.97/72.41
ACTIVE(g(0)) → c1 212.97/72.41
ACTIVE(f(g(0))) → c3(ACTIVE(g(0)))

212.97/72.41

(56) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0)))) 212.97/72.41
active(g(0)) → mark(s(0)) 212.97/72.41
active(g(s(z0))) → mark(s(s(g(z0)))) 212.97/72.41
active(sel(0, cons(z0, z1))) → mark(z0) 212.97/72.41
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 212.97/72.41
active(f(z0)) → f(active(z0)) 212.97/72.41
active(cons(z0, z1)) → cons(active(z0), z1) 212.97/72.41
active(g(z0)) → g(active(z0)) 212.97/72.41
active(s(z0)) → s(active(z0)) 212.97/72.41
active(sel(z0, z1)) → sel(active(z0), z1) 212.97/72.41
active(sel(z0, z1)) → sel(z0, active(z1)) 212.97/72.41
f(mark(z0)) → mark(f(z0)) 212.97/72.41
f(ok(z0)) → ok(f(z0)) 212.97/72.41
cons(mark(z0), z1) → mark(cons(z0, z1)) 212.97/72.41
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 212.97/72.41
g(mark(z0)) → mark(g(z0)) 212.97/72.41
g(ok(z0)) → ok(g(z0)) 212.97/72.41
s(mark(z0)) → mark(s(z0)) 212.97/72.41
s(ok(z0)) → ok(s(z0)) 212.97/72.41
sel(mark(z0), z1) → mark(sel(z0, z1)) 212.97/72.41
sel(z0, mark(z1)) → mark(sel(z0, z1)) 212.97/72.41
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 212.97/72.41
proper(f(z0)) → f(proper(z0)) 212.97/72.41
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 212.97/72.41
proper(g(z0)) → g(proper(z0)) 212.97/72.41
proper(0) → ok(0) 212.97/72.41
proper(s(z0)) → s(proper(z0)) 212.97/72.41
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 212.97/72.41
top(mark(z0)) → top(proper(z0)) 212.97/72.41
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.97/72.41
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.97/72.41
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.97/72.41
F(mark(z0)) → c11(F(z0)) 212.97/72.41
F(ok(z0)) → c12(F(z0)) 212.97/72.41
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.41
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.41
G(mark(z0)) → c15(G(z0)) 212.97/72.41
G(ok(z0)) → c16(G(z0)) 212.97/72.41
S(mark(z0)) → c17(S(z0)) 212.97/72.41
S(ok(z0)) → c18(S(z0)) 212.97/72.41
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.41
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.41
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.97/72.41
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.97/72.41
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.97/72.41
ACTIVE(f(z0)) → c(G(z0)) 212.97/72.41
ACTIVE(g(0)) → c1 212.97/72.41
ACTIVE(g(s(z0))) → c2(G(z0)) 212.97/72.41
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.97/72.41
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.97/72.41
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.97/72.41
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.97/72.41
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.97/72.41
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.97/72.41
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.97/72.41
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.97/72.41
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0))) 212.97/72.41
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0))) 212.97/72.41
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0))) 212.97/72.41
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(f(0)) → c22(F(ok(0))) 212.97/72.41
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.41
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.41
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.41
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.41
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.41
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.41
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.41
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.41
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0)) 212.97/72.41
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1)) 212.97/72.41
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0))) 212.97/72.41
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0))) 212.97/72.41
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0))) 212.97/72.41
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(g(0)) → c24(G(ok(0))) 212.97/72.41
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0))) 212.97/72.41
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0))) 212.97/72.41
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 212.97/72.41
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(s(0)) → c26(S(ok(0))) 212.97/72.41
PROPER(sel(x0, f(z0))) → c27(SEL(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.41
PROPER(sel(x0, cons(z0, z1))) → c27(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(sel(x0, g(z0))) → c27(SEL(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.41
PROPER(sel(x0, s(z0))) → c27(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.41
PROPER(sel(x0, sel(z0, z1))) → c27(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(sel(f(z0), x1)) → c27(SEL(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.41
PROPER(sel(cons(z0, z1), x1)) → c27(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.41
PROPER(sel(g(z0), x1)) → c27(SEL(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.41
PROPER(sel(s(z0), x1)) → c27(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.41
PROPER(sel(sel(z0, z1), x1)) → c27(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.41
PROPER(sel(x0, 0)) → c27(SEL(proper(x0), ok(0)), PROPER(x0)) 212.97/72.41
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)) 212.97/72.41
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.97/72.41
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.97/72.41
F(mark(z0)) → c11(F(z0)) 212.97/72.41
F(ok(z0)) → c12(F(z0)) 212.97/72.41
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.41
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.41
G(mark(z0)) → c15(G(z0)) 212.97/72.41
G(ok(z0)) → c16(G(z0)) 212.97/72.41
S(mark(z0)) → c17(S(z0)) 212.97/72.41
S(ok(z0)) → c18(S(z0)) 212.97/72.41
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.41
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.41
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.97/72.41
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 212.97/72.41
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.97/72.41
ACTIVE(f(z0)) → c(G(z0)) 212.97/72.41
ACTIVE(g(0)) → c1 212.97/72.41
ACTIVE(g(s(z0))) → c2(G(z0)) 212.97/72.41
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.97/72.41
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.97/72.41
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.97/72.41
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.97/72.41
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.97/72.41
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.97/72.41
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.97/72.41
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.97/72.41
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0))) 212.97/72.41
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0))) 212.97/72.41
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0))) 212.97/72.41
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(f(0)) → c22(F(ok(0))) 212.97/72.41
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.41
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.41
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.41
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.41
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.41
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.41
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.41
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.41
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0)) 212.97/72.41
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1)) 212.97/72.41
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0))) 212.97/72.41
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0))) 212.97/72.41
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0))) 212.97/72.41
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(g(0)) → c24(G(ok(0))) 212.97/72.41
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0))) 212.97/72.41
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0))) 212.97/72.41
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 212.97/72.41
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(s(0)) → c26(S(ok(0))) 212.97/72.41
PROPER(sel(x0, f(z0))) → c27(SEL(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.41
PROPER(sel(x0, cons(z0, z1))) → c27(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(sel(x0, g(z0))) → c27(SEL(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.41
PROPER(sel(x0, s(z0))) → c27(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.41
PROPER(sel(x0, sel(z0, z1))) → c27(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(sel(f(z0), x1)) → c27(SEL(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.41
PROPER(sel(cons(z0, z1), x1)) → c27(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.41
PROPER(sel(g(z0), x1)) → c27(SEL(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.41
PROPER(sel(s(z0), x1)) → c27(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.41
PROPER(sel(sel(z0, z1), x1)) → c27(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.41
PROPER(sel(x0, 0)) → c27(SEL(proper(x0), ok(0)), PROPER(x0)) 212.97/72.41
PROPER(sel(0, x1)) → c27(SEL(ok(0), proper(x1)), PROPER(x1))

K tuples:none
Defined Rule Symbols:

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

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, c1, c2, c5, c6, c7, c3, c8, c25, c22, c22, c23, c23, c24, c24, c26, c26, c27, c27

212.97/72.41

(57) 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))) 212.97/72.41
TOP(mark(cons(z0, z1))) → c28(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.41
TOP(mark(g(z0))) → c28(TOP(g(proper(z0))), PROPER(g(z0))) 212.97/72.41
TOP(mark(0)) → c28(TOP(ok(0)), PROPER(0)) 212.97/72.41
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0))) 212.97/72.41
TOP(mark(sel(z0, z1))) → c28(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))

212.97/72.41

(58) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0)))) 212.97/72.41
active(g(0)) → mark(s(0)) 212.97/72.41
active(g(s(z0))) → mark(s(s(g(z0)))) 212.97/72.41
active(sel(0, cons(z0, z1))) → mark(z0) 212.97/72.41
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 212.97/72.41
active(f(z0)) → f(active(z0)) 212.97/72.41
active(cons(z0, z1)) → cons(active(z0), z1) 212.97/72.41
active(g(z0)) → g(active(z0)) 212.97/72.41
active(s(z0)) → s(active(z0)) 212.97/72.41
active(sel(z0, z1)) → sel(active(z0), z1) 212.97/72.41
active(sel(z0, z1)) → sel(z0, active(z1)) 212.97/72.41
f(mark(z0)) → mark(f(z0)) 212.97/72.41
f(ok(z0)) → ok(f(z0)) 212.97/72.41
cons(mark(z0), z1) → mark(cons(z0, z1)) 212.97/72.41
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 212.97/72.41
g(mark(z0)) → mark(g(z0)) 212.97/72.41
g(ok(z0)) → ok(g(z0)) 212.97/72.41
s(mark(z0)) → mark(s(z0)) 212.97/72.41
s(ok(z0)) → ok(s(z0)) 212.97/72.41
sel(mark(z0), z1) → mark(sel(z0, z1)) 212.97/72.41
sel(z0, mark(z1)) → mark(sel(z0, z1)) 212.97/72.41
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 212.97/72.41
proper(f(z0)) → f(proper(z0)) 212.97/72.41
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 212.97/72.41
proper(g(z0)) → g(proper(z0)) 212.97/72.41
proper(0) → ok(0) 212.97/72.41
proper(s(z0)) → s(proper(z0)) 212.97/72.41
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 212.97/72.41
top(mark(z0)) → top(proper(z0)) 212.97/72.41
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.97/72.41
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.97/72.41
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.97/72.41
F(mark(z0)) → c11(F(z0)) 212.97/72.41
F(ok(z0)) → c12(F(z0)) 212.97/72.41
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.41
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.41
G(mark(z0)) → c15(G(z0)) 212.97/72.41
G(ok(z0)) → c16(G(z0)) 212.97/72.41
S(mark(z0)) → c17(S(z0)) 212.97/72.41
S(ok(z0)) → c18(S(z0)) 212.97/72.41
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.41
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.41
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.97/72.41
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.97/72.41
ACTIVE(f(z0)) → c(G(z0)) 212.97/72.41
ACTIVE(g(0)) → c1 212.97/72.41
ACTIVE(g(s(z0))) → c2(G(z0)) 212.97/72.41
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.97/72.41
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.97/72.41
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.97/72.41
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.97/72.41
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.97/72.41
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.97/72.41
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.41
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.41
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.41
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.97/72.41
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.41
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.97/72.41
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.97/72.41
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.41
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.97/72.41
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.97/72.41
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0))) 212.97/72.41
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0))) 212.97/72.41
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0))) 212.97/72.41
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(f(0)) → c22(F(ok(0))) 212.97/72.41
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.41
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.41
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.41
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.41
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.41
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.41
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.41
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.41
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0)) 212.97/72.41
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1)) 212.97/72.41
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0))) 212.97/72.41
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0))) 212.97/72.41
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0))) 212.97/72.41
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(g(0)) → c24(G(ok(0))) 212.97/72.41
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0))) 212.97/72.41
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0))) 212.97/72.41
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 212.97/72.41
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(s(0)) → c26(S(ok(0))) 212.97/72.41
PROPER(sel(x0, f(z0))) → c27(SEL(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.41
PROPER(sel(x0, cons(z0, z1))) → c27(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.41
PROPER(sel(x0, g(z0))) → c27(SEL(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.41
PROPER(sel(x0, s(z0))) → c27(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.41
PROPER(sel(x0, sel(z0, z1))) → c27(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.41
PROPER(sel(f(z0), x1)) → c27(SEL(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.41
PROPER(sel(cons(z0, z1), x1)) → c27(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.41
PROPER(sel(g(z0), x1)) → c27(SEL(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.41
PROPER(sel(s(z0), x1)) → c27(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.41
PROPER(sel(sel(z0, z1), x1)) → c27(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.41
PROPER(sel(x0, 0)) → c27(SEL(proper(x0), ok(0)), PROPER(x0)) 212.97/72.41
PROPER(sel(0, x1)) → c27(SEL(ok(0), proper(x1)), PROPER(x1)) 212.97/72.41
TOP(mark(f(z0))) → c28(TOP(f(proper(z0))), PROPER(f(z0))) 212.97/72.41
TOP(mark(cons(z0, z1))) → c28(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.41
TOP(mark(g(z0))) → c28(TOP(g(proper(z0))), PROPER(g(z0))) 212.97/72.41
TOP(mark(0)) → c28(TOP(ok(0)), PROPER(0)) 212.97/72.41
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0))) 212.97/72.41
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)) 212.97/72.41
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.97/72.41
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.97/72.41
F(mark(z0)) → c11(F(z0)) 212.97/72.41
F(ok(z0)) → c12(F(z0)) 212.97/72.42
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.42
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.42
G(mark(z0)) → c15(G(z0)) 212.97/72.42
G(ok(z0)) → c16(G(z0)) 212.97/72.42
S(mark(z0)) → c17(S(z0)) 212.97/72.42
S(ok(z0)) → c18(S(z0)) 212.97/72.42
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.42
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.42
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.97/72.42
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.97/72.42
ACTIVE(f(z0)) → c(G(z0)) 212.97/72.42
ACTIVE(g(0)) → c1 212.97/72.42
ACTIVE(g(s(z0))) → c2(G(z0)) 212.97/72.42
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.97/72.42
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.97/72.42
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.97/72.42
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.97/72.42
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.97/72.42
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.97/72.42
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.97/72.42
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.97/72.42
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(f(0)) → c22(F(ok(0))) 212.97/72.42
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.42
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.42
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.42
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.42
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.42
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.42
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0)) 212.97/72.42
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1)) 212.97/72.42
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(g(0)) → c24(G(ok(0))) 212.97/72.42
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(s(0)) → c26(S(ok(0))) 212.97/72.42
PROPER(sel(x0, f(z0))) → c27(SEL(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.42
PROPER(sel(x0, cons(z0, z1))) → c27(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(sel(x0, g(z0))) → c27(SEL(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.42
PROPER(sel(x0, s(z0))) → c27(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.42
PROPER(sel(x0, sel(z0, z1))) → c27(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(sel(f(z0), x1)) → c27(SEL(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.42
PROPER(sel(cons(z0, z1), x1)) → c27(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(sel(g(z0), x1)) → c27(SEL(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.42
PROPER(sel(s(z0), x1)) → c27(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.42
PROPER(sel(sel(z0, z1), x1)) → c27(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(sel(x0, 0)) → c27(SEL(proper(x0), ok(0)), PROPER(x0)) 212.97/72.42
PROPER(sel(0, x1)) → c27(SEL(ok(0), proper(x1)), PROPER(x1)) 212.97/72.42
TOP(mark(f(z0))) → c28(TOP(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
TOP(mark(cons(z0, z1))) → c28(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
TOP(mark(g(z0))) → c28(TOP(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
TOP(mark(0)) → c28(TOP(ok(0)), PROPER(0)) 212.97/72.42
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
TOP(mark(sel(z0, z1))) → c28(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1)))

K tuples:none
Defined Rule Symbols:

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

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, c1, c2, c5, c6, c7, c3, c8, c25, c22, c22, c23, c23, c24, c24, c26, c26, c27, c27, c28

212.97/72.42

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

Removed 1 trailing tuple parts

212.97/72.42

(60) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0)))) 212.97/72.42
active(g(0)) → mark(s(0)) 212.97/72.42
active(g(s(z0))) → mark(s(s(g(z0)))) 212.97/72.42
active(sel(0, cons(z0, z1))) → mark(z0) 212.97/72.42
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 212.97/72.42
active(f(z0)) → f(active(z0)) 212.97/72.42
active(cons(z0, z1)) → cons(active(z0), z1) 212.97/72.42
active(g(z0)) → g(active(z0)) 212.97/72.42
active(s(z0)) → s(active(z0)) 212.97/72.42
active(sel(z0, z1)) → sel(active(z0), z1) 212.97/72.42
active(sel(z0, z1)) → sel(z0, active(z1)) 212.97/72.42
f(mark(z0)) → mark(f(z0)) 212.97/72.42
f(ok(z0)) → ok(f(z0)) 212.97/72.42
cons(mark(z0), z1) → mark(cons(z0, z1)) 212.97/72.42
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 212.97/72.42
g(mark(z0)) → mark(g(z0)) 212.97/72.42
g(ok(z0)) → ok(g(z0)) 212.97/72.42
s(mark(z0)) → mark(s(z0)) 212.97/72.42
s(ok(z0)) → ok(s(z0)) 212.97/72.42
sel(mark(z0), z1) → mark(sel(z0, z1)) 212.97/72.42
sel(z0, mark(z1)) → mark(sel(z0, z1)) 212.97/72.42
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 212.97/72.42
proper(f(z0)) → f(proper(z0)) 212.97/72.42
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 212.97/72.42
proper(g(z0)) → g(proper(z0)) 212.97/72.42
proper(0) → ok(0) 212.97/72.42
proper(s(z0)) → s(proper(z0)) 212.97/72.42
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 212.97/72.42
top(mark(z0)) → top(proper(z0)) 212.97/72.42
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.97/72.42
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.97/72.42
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.97/72.42
F(mark(z0)) → c11(F(z0)) 212.97/72.42
F(ok(z0)) → c12(F(z0)) 212.97/72.42
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.42
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.42
G(mark(z0)) → c15(G(z0)) 212.97/72.42
G(ok(z0)) → c16(G(z0)) 212.97/72.42
S(mark(z0)) → c17(S(z0)) 212.97/72.42
S(ok(z0)) → c18(S(z0)) 212.97/72.42
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.42
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.42
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.97/72.42
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.97/72.42
ACTIVE(f(z0)) → c(G(z0)) 212.97/72.42
ACTIVE(g(0)) → c1 212.97/72.42
ACTIVE(g(s(z0))) → c2(G(z0)) 212.97/72.42
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.97/72.42
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.97/72.42
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.97/72.42
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.97/72.42
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.97/72.42
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.97/72.42
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.97/72.42
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.97/72.42
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(f(0)) → c22(F(ok(0))) 212.97/72.42
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.42
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.42
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.42
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.42
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.42
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.42
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0)) 212.97/72.42
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1)) 212.97/72.42
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(g(0)) → c24(G(ok(0))) 212.97/72.42
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(s(0)) → c26(S(ok(0))) 212.97/72.42
PROPER(sel(x0, f(z0))) → c27(SEL(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.42
PROPER(sel(x0, cons(z0, z1))) → c27(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(sel(x0, g(z0))) → c27(SEL(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.42
PROPER(sel(x0, s(z0))) → c27(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.42
PROPER(sel(x0, sel(z0, z1))) → c27(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(sel(f(z0), x1)) → c27(SEL(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.42
PROPER(sel(cons(z0, z1), x1)) → c27(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(sel(g(z0), x1)) → c27(SEL(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.42
PROPER(sel(s(z0), x1)) → c27(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.42
PROPER(sel(sel(z0, z1), x1)) → c27(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(sel(x0, 0)) → c27(SEL(proper(x0), ok(0)), PROPER(x0)) 212.97/72.42
PROPER(sel(0, x1)) → c27(SEL(ok(0), proper(x1)), PROPER(x1)) 212.97/72.42
TOP(mark(f(z0))) → c28(TOP(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
TOP(mark(cons(z0, z1))) → c28(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
TOP(mark(g(z0))) → c28(TOP(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
TOP(mark(sel(z0, z1))) → c28(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.42
TOP(mark(0)) → c28(TOP(ok(0)))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.97/72.42
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.97/72.42
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.97/72.42
F(mark(z0)) → c11(F(z0)) 212.97/72.42
F(ok(z0)) → c12(F(z0)) 212.97/72.42
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.42
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.42
G(mark(z0)) → c15(G(z0)) 212.97/72.42
G(ok(z0)) → c16(G(z0)) 212.97/72.42
S(mark(z0)) → c17(S(z0)) 212.97/72.42
S(ok(z0)) → c18(S(z0)) 212.97/72.42
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.42
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.42
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.97/72.42
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.97/72.42
ACTIVE(f(z0)) → c(G(z0)) 212.97/72.42
ACTIVE(g(0)) → c1 212.97/72.42
ACTIVE(g(s(z0))) → c2(G(z0)) 212.97/72.42
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.97/72.42
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.97/72.42
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.97/72.42
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.97/72.42
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.97/72.42
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.97/72.42
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.97/72.42
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.97/72.42
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(f(0)) → c22(F(ok(0))) 212.97/72.42
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.42
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.42
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.42
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.42
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.42
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.42
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0)) 212.97/72.42
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1)) 212.97/72.42
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(g(0)) → c24(G(ok(0))) 212.97/72.42
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(s(0)) → c26(S(ok(0))) 212.97/72.42
PROPER(sel(x0, f(z0))) → c27(SEL(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.42
PROPER(sel(x0, cons(z0, z1))) → c27(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(sel(x0, g(z0))) → c27(SEL(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.42
PROPER(sel(x0, s(z0))) → c27(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.42
PROPER(sel(x0, sel(z0, z1))) → c27(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(sel(f(z0), x1)) → c27(SEL(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.42
PROPER(sel(cons(z0, z1), x1)) → c27(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(sel(g(z0), x1)) → c27(SEL(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.42
PROPER(sel(s(z0), x1)) → c27(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.42
PROPER(sel(sel(z0, z1), x1)) → c27(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(sel(x0, 0)) → c27(SEL(proper(x0), ok(0)), PROPER(x0)) 212.97/72.42
PROPER(sel(0, x1)) → c27(SEL(ok(0), proper(x1)), PROPER(x1)) 212.97/72.42
TOP(mark(f(z0))) → c28(TOP(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
TOP(mark(cons(z0, z1))) → c28(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
TOP(mark(g(z0))) → c28(TOP(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
TOP(mark(sel(z0, z1))) → c28(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.42
TOP(mark(0)) → c28(TOP(ok(0)))

K tuples:none
Defined Rule Symbols:

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

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, c1, c2, c5, c6, c7, c3, c8, c25, c22, c22, c23, c23, c24, c24, c26, c26, c27, c27, c28, c28

212.97/72.42

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

Removed 5 trailing nodes:

ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.97/72.42
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.97/72.42
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.97/72.42
ACTIVE(g(0)) → c1 212.97/72.42
ACTIVE(f(g(0))) → c3(ACTIVE(g(0)))

212.97/72.42

(62) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0)))) 212.97/72.42
active(g(0)) → mark(s(0)) 212.97/72.42
active(g(s(z0))) → mark(s(s(g(z0)))) 212.97/72.42
active(sel(0, cons(z0, z1))) → mark(z0) 212.97/72.42
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 212.97/72.42
active(f(z0)) → f(active(z0)) 212.97/72.42
active(cons(z0, z1)) → cons(active(z0), z1) 212.97/72.42
active(g(z0)) → g(active(z0)) 212.97/72.42
active(s(z0)) → s(active(z0)) 212.97/72.42
active(sel(z0, z1)) → sel(active(z0), z1) 212.97/72.42
active(sel(z0, z1)) → sel(z0, active(z1)) 212.97/72.42
f(mark(z0)) → mark(f(z0)) 212.97/72.42
f(ok(z0)) → ok(f(z0)) 212.97/72.42
cons(mark(z0), z1) → mark(cons(z0, z1)) 212.97/72.42
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 212.97/72.42
g(mark(z0)) → mark(g(z0)) 212.97/72.42
g(ok(z0)) → ok(g(z0)) 212.97/72.42
s(mark(z0)) → mark(s(z0)) 212.97/72.42
s(ok(z0)) → ok(s(z0)) 212.97/72.42
sel(mark(z0), z1) → mark(sel(z0, z1)) 212.97/72.42
sel(z0, mark(z1)) → mark(sel(z0, z1)) 212.97/72.42
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 212.97/72.42
proper(f(z0)) → f(proper(z0)) 212.97/72.42
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 212.97/72.42
proper(g(z0)) → g(proper(z0)) 212.97/72.42
proper(0) → ok(0) 212.97/72.42
proper(s(z0)) → s(proper(z0)) 212.97/72.42
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 212.97/72.42
top(mark(z0)) → top(proper(z0)) 212.97/72.42
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.97/72.42
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.97/72.42
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.97/72.42
F(mark(z0)) → c11(F(z0)) 212.97/72.42
F(ok(z0)) → c12(F(z0)) 212.97/72.42
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.42
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.42
G(mark(z0)) → c15(G(z0)) 212.97/72.42
G(ok(z0)) → c16(G(z0)) 212.97/72.42
S(mark(z0)) → c17(S(z0)) 212.97/72.42
S(ok(z0)) → c18(S(z0)) 212.97/72.42
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.42
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.42
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.97/72.42
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.97/72.42
ACTIVE(f(z0)) → c(G(z0)) 212.97/72.42
ACTIVE(g(0)) → c1 212.97/72.42
ACTIVE(g(s(z0))) → c2(G(z0)) 212.97/72.42
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.97/72.42
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.97/72.42
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.97/72.42
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.97/72.42
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.97/72.42
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.97/72.42
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.97/72.42
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.97/72.42
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(f(0)) → c22(F(ok(0))) 212.97/72.42
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.42
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.42
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.42
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.42
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.42
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.42
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0)) 212.97/72.42
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1)) 212.97/72.42
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(g(0)) → c24(G(ok(0))) 212.97/72.42
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(s(0)) → c26(S(ok(0))) 212.97/72.42
PROPER(sel(x0, f(z0))) → c27(SEL(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.42
PROPER(sel(x0, cons(z0, z1))) → c27(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(sel(x0, g(z0))) → c27(SEL(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.42
PROPER(sel(x0, s(z0))) → c27(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.42
PROPER(sel(x0, sel(z0, z1))) → c27(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(sel(f(z0), x1)) → c27(SEL(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.42
PROPER(sel(cons(z0, z1), x1)) → c27(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(sel(g(z0), x1)) → c27(SEL(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.42
PROPER(sel(s(z0), x1)) → c27(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.42
PROPER(sel(sel(z0, z1), x1)) → c27(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(sel(x0, 0)) → c27(SEL(proper(x0), ok(0)), PROPER(x0)) 212.97/72.42
PROPER(sel(0, x1)) → c27(SEL(ok(0), proper(x1)), PROPER(x1)) 212.97/72.42
TOP(mark(f(z0))) → c28(TOP(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
TOP(mark(cons(z0, z1))) → c28(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
TOP(mark(g(z0))) → c28(TOP(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
TOP(mark(sel(z0, z1))) → c28(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.42
TOP(mark(0)) → c28(TOP(ok(0)))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.97/72.42
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.97/72.42
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.97/72.42
F(mark(z0)) → c11(F(z0)) 212.97/72.42
F(ok(z0)) → c12(F(z0)) 212.97/72.42
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.42
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.42
G(mark(z0)) → c15(G(z0)) 212.97/72.42
G(ok(z0)) → c16(G(z0)) 212.97/72.42
S(mark(z0)) → c17(S(z0)) 212.97/72.42
S(ok(z0)) → c18(S(z0)) 212.97/72.42
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.42
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.42
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.97/72.42
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.97/72.42
ACTIVE(f(z0)) → c(G(z0)) 212.97/72.42
ACTIVE(g(0)) → c1 212.97/72.42
ACTIVE(g(s(z0))) → c2(G(z0)) 212.97/72.42
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.97/72.42
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.97/72.42
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.97/72.42
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.97/72.42
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.97/72.42
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.97/72.42
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.97/72.42
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.97/72.42
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(f(0)) → c22(F(ok(0))) 212.97/72.42
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.42
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.42
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.42
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.42
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.42
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.42
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0)) 212.97/72.42
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1)) 212.97/72.42
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(g(0)) → c24(G(ok(0))) 212.97/72.42
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(s(0)) → c26(S(ok(0))) 212.97/72.42
PROPER(sel(x0, f(z0))) → c27(SEL(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.42
PROPER(sel(x0, cons(z0, z1))) → c27(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(sel(x0, g(z0))) → c27(SEL(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.42
PROPER(sel(x0, s(z0))) → c27(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.42
PROPER(sel(x0, sel(z0, z1))) → c27(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(sel(f(z0), x1)) → c27(SEL(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.42
PROPER(sel(cons(z0, z1), x1)) → c27(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(sel(g(z0), x1)) → c27(SEL(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.42
PROPER(sel(s(z0), x1)) → c27(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.42
PROPER(sel(sel(z0, z1), x1)) → c27(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(sel(x0, 0)) → c27(SEL(proper(x0), ok(0)), PROPER(x0)) 212.97/72.42
PROPER(sel(0, x1)) → c27(SEL(ok(0), proper(x1)), PROPER(x1)) 212.97/72.42
TOP(mark(f(z0))) → c28(TOP(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
TOP(mark(cons(z0, z1))) → c28(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
TOP(mark(g(z0))) → c28(TOP(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
TOP(mark(sel(z0, z1))) → c28(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.42
TOP(mark(0)) → c28(TOP(ok(0)))

K tuples:none
Defined Rule Symbols:

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

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, c1, c2, c5, c6, c7, c3, c8, c25, c22, c22, c23, c23, c24, c24, c26, c26, c27, c27, c28, c28

212.97/72.42

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

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

TOP(mark(0)) → c28(TOP(ok(0)))

We considered the (Usable) Rules:

proper(f(z0)) → f(proper(z0)) 212.97/72.42
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 212.97/72.42
proper(g(z0)) → g(proper(z0)) 212.97/72.42
proper(0) → ok(0) 212.97/72.42
proper(s(z0)) → s(proper(z0)) 212.97/72.42
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 212.97/72.42
sel(mark(z0), z1) → mark(sel(z0, z1)) 212.97/72.42
sel(z0, mark(z1)) → mark(sel(z0, z1)) 212.97/72.42
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 212.97/72.42
s(mark(z0)) → mark(s(z0)) 212.97/72.42
s(ok(z0)) → ok(s(z0)) 212.97/72.42
g(mark(z0)) → mark(g(z0)) 212.97/72.42
g(ok(z0)) → ok(g(z0)) 212.97/72.42
cons(mark(z0), z1) → mark(cons(z0, z1)) 212.97/72.42
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 212.97/72.42
f(mark(z0)) → mark(f(z0)) 212.97/72.42
f(ok(z0)) → ok(f(z0)) 212.97/72.42
active(f(z0)) → mark(cons(z0, f(g(z0)))) 212.97/72.42
active(g(0)) → mark(s(0)) 212.97/72.42
active(g(s(z0))) → mark(s(s(g(z0)))) 212.97/72.42
active(sel(0, cons(z0, z1))) → mark(z0) 212.97/72.42
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 212.97/72.42
active(f(z0)) → f(active(z0)) 212.97/72.42
active(cons(z0, z1)) → cons(active(z0), z1) 212.97/72.42
active(g(z0)) → g(active(z0)) 212.97/72.42
active(s(z0)) → s(active(z0)) 212.97/72.42
active(sel(z0, z1)) → sel(active(z0), z1) 212.97/72.42
active(sel(z0, z1)) → sel(z0, active(z1))

And the Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.97/72.42
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.97/72.42
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.97/72.42
F(mark(z0)) → c11(F(z0)) 212.97/72.42
F(ok(z0)) → c12(F(z0)) 212.97/72.42
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.42
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.42
G(mark(z0)) → c15(G(z0)) 212.97/72.42
G(ok(z0)) → c16(G(z0)) 212.97/72.42
S(mark(z0)) → c17(S(z0)) 212.97/72.42
S(ok(z0)) → c18(S(z0)) 212.97/72.42
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.42
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.42
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.97/72.42
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.97/72.42
ACTIVE(f(z0)) → c(G(z0)) 212.97/72.42
ACTIVE(g(0)) → c1 212.97/72.42
ACTIVE(g(s(z0))) → c2(G(z0)) 212.97/72.42
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.97/72.42
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.97/72.42
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.97/72.42
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.97/72.42
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.97/72.42
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.97/72.42
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.97/72.42
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.97/72.42
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(f(0)) → c22(F(ok(0))) 212.97/72.42
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.42
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.42
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.42
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.42
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.42
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.42
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0)) 212.97/72.42
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1)) 212.97/72.42
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(g(0)) → c24(G(ok(0))) 212.97/72.42
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(s(0)) → c26(S(ok(0))) 212.97/72.42
PROPER(sel(x0, f(z0))) → c27(SEL(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.42
PROPER(sel(x0, cons(z0, z1))) → c27(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(sel(x0, g(z0))) → c27(SEL(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.42
PROPER(sel(x0, s(z0))) → c27(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.42
PROPER(sel(x0, sel(z0, z1))) → c27(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(sel(f(z0), x1)) → c27(SEL(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.42
PROPER(sel(cons(z0, z1), x1)) → c27(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(sel(g(z0), x1)) → c27(SEL(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.42
PROPER(sel(s(z0), x1)) → c27(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.42
PROPER(sel(sel(z0, z1), x1)) → c27(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(sel(x0, 0)) → c27(SEL(proper(x0), ok(0)), PROPER(x0)) 212.97/72.42
PROPER(sel(0, x1)) → c27(SEL(ok(0), proper(x1)), PROPER(x1)) 212.97/72.42
TOP(mark(f(z0))) → c28(TOP(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
TOP(mark(cons(z0, z1))) → c28(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
TOP(mark(g(z0))) → c28(TOP(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
TOP(mark(sel(z0, z1))) → c28(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.42
TOP(mark(0)) → c28(TOP(ok(0)))

The order we found is given by the following interpretation:
Polynomial interpretation : 212.97/72.42

POL(0) = 0 212.97/72.42
POL(ACTIVE(x₁)) = 0 212.97/72.42
POL(CONS(x₁, x₂)) = 0 212.97/72.42
POL(F(x₁)) = 0 212.97/72.42
POL(G(x₁)) = 0 212.97/72.42
POL(PROPER(x₁)) = 0 212.97/72.42
POL(S(x₁)) = 0 212.97/72.42
POL(SEL(x₁, x₂)) = 0 212.97/72.42
POL(TOP(x₁)) = x₁ 212.97/72.42
POL(active(x₁)) = x₁ 212.97/72.42
POL(c(x₁)) = x₁ 212.97/72.42
POL(c1) = 0 212.97/72.42
POL(c10(x₁, x₂)) = x₁ + x₂ 212.97/72.42
POL(c11(x₁)) = x₁ 212.97/72.42
POL(c12(x₁)) = x₁ 212.97/72.42
POL(c13(x₁)) = x₁ 212.97/72.42
POL(c14(x₁)) = x₁ 212.97/72.42
POL(c15(x₁)) = x₁ 212.97/72.42
POL(c16(x₁)) = x₁ 212.97/72.42
POL(c17(x₁)) = x₁ 212.97/72.42
POL(c18(x₁)) = x₁ 212.97/72.42
POL(c19(x₁)) = x₁ 212.97/72.42
POL(c2(x₁)) = x₁ 212.97/72.42
POL(c20(x₁)) = x₁ 212.97/72.42
POL(c21(x₁)) = x₁ 212.97/72.42
POL(c22(x₁)) = x₁ 212.97/72.42
POL(c22(x₁, x₂)) = x₁ + x₂ 212.97/72.42
POL(c23(x₁, x₂)) = x₁ + x₂ 212.97/72.42
POL(c23(x₁, x₂, x₃)) = x₁ + x₂ + x₃ 212.97/72.42
POL(c24(x₁)) = x₁ 212.97/72.42
POL(c24(x₁, x₂)) = x₁ + x₂ 212.97/72.42
POL(c25(x₁)) = x₁ 212.97/72.42
POL(c26(x₁)) = x₁ 212.97/72.42
POL(c26(x₁, x₂)) = x₁ + x₂ 212.97/72.42
POL(c27(x₁, x₂)) = x₁ + x₂ 212.97/72.42
POL(c27(x₁, x₂, x₃)) = x₁ + x₂ + x₃ 212.97/72.42
POL(c28(x₁)) = x₁ 212.97/72.42
POL(c28(x₁, x₂)) = x₁ + x₂ 212.97/72.42
POL(c29(x₁, x₂)) = x₁ + x₂ 212.97/72.42
POL(c3(x₁)) = x₁ 212.97/72.42
POL(c4(x₁)) = x₁ 212.97/72.42
POL(c5(x₁, x₂)) = x₁ + x₂ 212.97/72.42
POL(c6(x₁, x₂)) = x₁ + x₂ 212.97/72.42
POL(c7(x₁, x₂)) = x₁ + x₂ 212.97/72.42
POL(c8(x₁, x₂)) = x₁ + x₂ 212.97/72.42
POL(c9(x₁, x₂)) = x₁ + x₂ 212.97/72.42
POL(cons(x₁, x₂)) = [4] 212.97/72.42
POL(f(x₁)) = [4] 212.97/72.42
POL(g(x₁)) = [4] 212.97/72.42
POL(mark(x₁)) = [4] 212.97/72.42
POL(ok(x₁)) = x₁ 212.97/72.42
POL(proper(x₁)) = 0 212.97/72.42
POL(s(x₁)) = [4] 212.97/72.42
POL(sel(x₁, x₂)) = [4]

212.97/72.42

(64) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0)))) 212.97/72.42
active(g(0)) → mark(s(0)) 212.97/72.42
active(g(s(z0))) → mark(s(s(g(z0)))) 212.97/72.42
active(sel(0, cons(z0, z1))) → mark(z0) 212.97/72.42
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 212.97/72.42
active(f(z0)) → f(active(z0)) 212.97/72.42
active(cons(z0, z1)) → cons(active(z0), z1) 212.97/72.42
active(g(z0)) → g(active(z0)) 212.97/72.42
active(s(z0)) → s(active(z0)) 212.97/72.42
active(sel(z0, z1)) → sel(active(z0), z1) 212.97/72.42
active(sel(z0, z1)) → sel(z0, active(z1)) 212.97/72.42
f(mark(z0)) → mark(f(z0)) 212.97/72.42
f(ok(z0)) → ok(f(z0)) 212.97/72.42
cons(mark(z0), z1) → mark(cons(z0, z1)) 212.97/72.42
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 212.97/72.42
g(mark(z0)) → mark(g(z0)) 212.97/72.42
g(ok(z0)) → ok(g(z0)) 212.97/72.42
s(mark(z0)) → mark(s(z0)) 212.97/72.42
s(ok(z0)) → ok(s(z0)) 212.97/72.42
sel(mark(z0), z1) → mark(sel(z0, z1)) 212.97/72.42
sel(z0, mark(z1)) → mark(sel(z0, z1)) 212.97/72.42
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 212.97/72.42
proper(f(z0)) → f(proper(z0)) 212.97/72.42
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 212.97/72.42
proper(g(z0)) → g(proper(z0)) 212.97/72.42
proper(0) → ok(0) 212.97/72.42
proper(s(z0)) → s(proper(z0)) 212.97/72.42
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 212.97/72.42
top(mark(z0)) → top(proper(z0)) 212.97/72.42
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.97/72.42
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.97/72.42
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.97/72.42
F(mark(z0)) → c11(F(z0)) 212.97/72.42
F(ok(z0)) → c12(F(z0)) 212.97/72.42
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.42
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.42
G(mark(z0)) → c15(G(z0)) 212.97/72.42
G(ok(z0)) → c16(G(z0)) 212.97/72.42
S(mark(z0)) → c17(S(z0)) 212.97/72.42
S(ok(z0)) → c18(S(z0)) 212.97/72.42
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.42
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.42
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.97/72.42
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.97/72.42
ACTIVE(f(z0)) → c(G(z0)) 212.97/72.42
ACTIVE(g(0)) → c1 212.97/72.42
ACTIVE(g(s(z0))) → c2(G(z0)) 212.97/72.42
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.97/72.42
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.97/72.42
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.97/72.42
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.97/72.42
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.97/72.42
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.97/72.42
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.97/72.42
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.97/72.42
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(f(0)) → c22(F(ok(0))) 212.97/72.42
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.42
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.42
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.42
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.42
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.42
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.42
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0)) 212.97/72.42
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1)) 212.97/72.42
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(g(0)) → c24(G(ok(0))) 212.97/72.42
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(s(0)) → c26(S(ok(0))) 212.97/72.42
PROPER(sel(x0, f(z0))) → c27(SEL(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.42
PROPER(sel(x0, cons(z0, z1))) → c27(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(sel(x0, g(z0))) → c27(SEL(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.42
PROPER(sel(x0, s(z0))) → c27(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.42
PROPER(sel(x0, sel(z0, z1))) → c27(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(sel(f(z0), x1)) → c27(SEL(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.42
PROPER(sel(cons(z0, z1), x1)) → c27(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(sel(g(z0), x1)) → c27(SEL(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.42
PROPER(sel(s(z0), x1)) → c27(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.42
PROPER(sel(sel(z0, z1), x1)) → c27(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(sel(x0, 0)) → c27(SEL(proper(x0), ok(0)), PROPER(x0)) 212.97/72.42
PROPER(sel(0, x1)) → c27(SEL(ok(0), proper(x1)), PROPER(x1)) 212.97/72.42
TOP(mark(f(z0))) → c28(TOP(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
TOP(mark(cons(z0, z1))) → c28(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
TOP(mark(g(z0))) → c28(TOP(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
TOP(mark(sel(z0, z1))) → c28(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.42
TOP(mark(0)) → c28(TOP(ok(0)))

S tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.97/72.42
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.97/72.42
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.97/72.42
F(mark(z0)) → c11(F(z0)) 212.97/72.42
F(ok(z0)) → c12(F(z0)) 212.97/72.42
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.42
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.42
G(mark(z0)) → c15(G(z0)) 212.97/72.42
G(ok(z0)) → c16(G(z0)) 212.97/72.42
S(mark(z0)) → c17(S(z0)) 212.97/72.42
S(ok(z0)) → c18(S(z0)) 212.97/72.42
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.42
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.42
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.97/72.42
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 212.97/72.42
ACTIVE(f(z0)) → c(G(z0)) 212.97/72.42
ACTIVE(g(0)) → c1 212.97/72.42
ACTIVE(g(s(z0))) → c2(G(z0)) 212.97/72.42
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.97/72.42
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.97/72.42
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.97/72.42
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.97/72.42
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.97/72.42
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.97/72.42
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.97/72.42
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.97/72.42
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(f(0)) → c22(F(ok(0))) 212.97/72.42
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.42
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.42
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.42
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.42
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.42
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.42
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0)) 212.97/72.42
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1)) 212.97/72.42
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(g(0)) → c24(G(ok(0))) 212.97/72.42
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(s(0)) → c26(S(ok(0))) 212.97/72.42
PROPER(sel(x0, f(z0))) → c27(SEL(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.42
PROPER(sel(x0, cons(z0, z1))) → c27(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(sel(x0, g(z0))) → c27(SEL(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.42
PROPER(sel(x0, s(z0))) → c27(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.42
PROPER(sel(x0, sel(z0, z1))) → c27(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(sel(f(z0), x1)) → c27(SEL(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.42
PROPER(sel(cons(z0, z1), x1)) → c27(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(sel(g(z0), x1)) → c27(SEL(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.42
PROPER(sel(s(z0), x1)) → c27(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.42
PROPER(sel(sel(z0, z1), x1)) → c27(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(sel(x0, 0)) → c27(SEL(proper(x0), ok(0)), PROPER(x0)) 212.97/72.42
PROPER(sel(0, x1)) → c27(SEL(ok(0), proper(x1)), PROPER(x1)) 212.97/72.42
TOP(mark(f(z0))) → c28(TOP(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
TOP(mark(cons(z0, z1))) → c28(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
TOP(mark(g(z0))) → c28(TOP(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
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, f, cons, g, s, sel, proper, top

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, c1, c2, c5, c6, c7, c3, c8, c25, c22, c22, c23, c23, c24, c24, c26, c26, c27, c27, c28, c28

212.97/72.42

(65) 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))) 212.97/72.42
TOP(ok(g(0))) → c29(TOP(mark(s(0))), ACTIVE(g(0))) 212.97/72.42
TOP(ok(g(s(z0)))) → c29(TOP(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.42
TOP(ok(sel(0, cons(z0, z1)))) → c29(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
TOP(ok(sel(s(z0), cons(z1, z2)))) → c29(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
TOP(ok(f(z0))) → c29(TOP(f(active(z0))), ACTIVE(f(z0))) 212.97/72.42
TOP(ok(cons(z0, z1))) → c29(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.42
TOP(ok(g(z0))) → c29(TOP(g(active(z0))), ACTIVE(g(z0))) 212.97/72.42
TOP(ok(s(z0))) → c29(TOP(s(active(z0))), ACTIVE(s(z0))) 212.97/72.42
TOP(ok(sel(z0, z1))) → c29(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.42
TOP(ok(sel(z0, z1))) → c29(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))

212.97/72.42

(66) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0)))) 212.97/72.42
active(g(0)) → mark(s(0)) 212.97/72.42
active(g(s(z0))) → mark(s(s(g(z0)))) 212.97/72.42
active(sel(0, cons(z0, z1))) → mark(z0) 212.97/72.42
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 212.97/72.42
active(f(z0)) → f(active(z0)) 212.97/72.42
active(cons(z0, z1)) → cons(active(z0), z1) 212.97/72.42
active(g(z0)) → g(active(z0)) 212.97/72.42
active(s(z0)) → s(active(z0)) 212.97/72.42
active(sel(z0, z1)) → sel(active(z0), z1) 212.97/72.42
active(sel(z0, z1)) → sel(z0, active(z1)) 212.97/72.42
f(mark(z0)) → mark(f(z0)) 212.97/72.42
f(ok(z0)) → ok(f(z0)) 212.97/72.42
cons(mark(z0), z1) → mark(cons(z0, z1)) 212.97/72.42
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 212.97/72.42
g(mark(z0)) → mark(g(z0)) 212.97/72.42
g(ok(z0)) → ok(g(z0)) 212.97/72.42
s(mark(z0)) → mark(s(z0)) 212.97/72.42
s(ok(z0)) → ok(s(z0)) 212.97/72.42
sel(mark(z0), z1) → mark(sel(z0, z1)) 212.97/72.42
sel(z0, mark(z1)) → mark(sel(z0, z1)) 212.97/72.42
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 212.97/72.42
proper(f(z0)) → f(proper(z0)) 212.97/72.42
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 212.97/72.42
proper(g(z0)) → g(proper(z0)) 212.97/72.42
proper(0) → ok(0) 212.97/72.42
proper(s(z0)) → s(proper(z0)) 212.97/72.42
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 212.97/72.42
top(mark(z0)) → top(proper(z0)) 212.97/72.42
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(sel(s(z0), cons(z1, z2))) → c4(SEL(z0, z2)) 212.97/72.42
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.97/72.42
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.97/72.42
F(mark(z0)) → c11(F(z0)) 212.97/72.42
F(ok(z0)) → c12(F(z0)) 212.97/72.42
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.42
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.42
G(mark(z0)) → c15(G(z0)) 212.97/72.42
G(ok(z0)) → c16(G(z0)) 212.97/72.42
S(mark(z0)) → c17(S(z0)) 212.97/72.42
S(ok(z0)) → c18(S(z0)) 212.97/72.42
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.42
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.42
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.97/72.42
ACTIVE(f(z0)) → c(G(z0)) 212.97/72.42
ACTIVE(g(0)) → c1 212.97/72.42
ACTIVE(g(s(z0))) → c2(G(z0)) 212.97/72.42
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.97/72.42
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.97/72.42
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.97/72.42
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.97/72.42
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.97/72.42
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.97/72.42
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.97/72.42
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.97/72.42
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(f(0)) → c22(F(ok(0))) 212.97/72.42
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.42
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.42
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.42
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.42
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.42
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.42
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0)) 212.97/72.42
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1)) 212.97/72.42
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(g(0)) → c24(G(ok(0))) 212.97/72.42
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(s(0)) → c26(S(ok(0))) 212.97/72.42
PROPER(sel(x0, f(z0))) → c27(SEL(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.42
PROPER(sel(x0, cons(z0, z1))) → c27(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(sel(x0, g(z0))) → c27(SEL(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.42
PROPER(sel(x0, s(z0))) → c27(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.42
PROPER(sel(x0, sel(z0, z1))) → c27(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(sel(f(z0), x1)) → c27(SEL(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.42
PROPER(sel(cons(z0, z1), x1)) → c27(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(sel(g(z0), x1)) → c27(SEL(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.42
PROPER(sel(s(z0), x1)) → c27(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.42
PROPER(sel(sel(z0, z1), x1)) → c27(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(sel(x0, 0)) → c27(SEL(proper(x0), ok(0)), PROPER(x0)) 212.97/72.42
PROPER(sel(0, x1)) → c27(SEL(ok(0), proper(x1)), PROPER(x1)) 212.97/72.42
TOP(mark(f(z0))) → c28(TOP(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
TOP(mark(cons(z0, z1))) → c28(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
TOP(mark(g(z0))) → c28(TOP(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
TOP(mark(sel(z0, z1))) → c28(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.42
TOP(mark(0)) → c28(TOP(ok(0))) 212.97/72.42
TOP(ok(f(z0))) → c29(TOP(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.42
TOP(ok(g(0))) → c29(TOP(mark(s(0))), ACTIVE(g(0))) 212.97/72.42
TOP(ok(g(s(z0)))) → c29(TOP(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.42
TOP(ok(sel(0, cons(z0, z1)))) → c29(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
TOP(ok(sel(s(z0), cons(z1, z2)))) → c29(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
TOP(ok(f(z0))) → c29(TOP(f(active(z0))), ACTIVE(f(z0))) 212.97/72.42
TOP(ok(cons(z0, z1))) → c29(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.42
TOP(ok(g(z0))) → c29(TOP(g(active(z0))), ACTIVE(g(z0))) 212.97/72.42
TOP(ok(s(z0))) → c29(TOP(s(active(z0))), ACTIVE(s(z0))) 212.97/72.42
TOP(ok(sel(z0, z1))) → c29(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.42
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)) 212.97/72.42
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.97/72.42
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.97/72.42
F(mark(z0)) → c11(F(z0)) 212.97/72.42
F(ok(z0)) → c12(F(z0)) 212.97/72.42
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.42
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.42
G(mark(z0)) → c15(G(z0)) 212.97/72.42
G(ok(z0)) → c16(G(z0)) 212.97/72.42
S(mark(z0)) → c17(S(z0)) 212.97/72.42
S(ok(z0)) → c18(S(z0)) 212.97/72.42
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.42
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.42
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.97/72.42
ACTIVE(f(z0)) → c(G(z0)) 212.97/72.42
ACTIVE(g(0)) → c1 212.97/72.42
ACTIVE(g(s(z0))) → c2(G(z0)) 212.97/72.42
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.97/72.42
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.97/72.42
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.97/72.42
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.97/72.42
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.97/72.42
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.97/72.42
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.97/72.42
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.97/72.42
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(f(0)) → c22(F(ok(0))) 212.97/72.42
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.42
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.42
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.42
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.42
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.42
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.42
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0)) 212.97/72.42
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1)) 212.97/72.42
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(g(0)) → c24(G(ok(0))) 212.97/72.42
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(s(0)) → c26(S(ok(0))) 212.97/72.42
PROPER(sel(x0, f(z0))) → c27(SEL(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.42
PROPER(sel(x0, cons(z0, z1))) → c27(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(sel(x0, g(z0))) → c27(SEL(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.42
PROPER(sel(x0, s(z0))) → c27(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.42
PROPER(sel(x0, sel(z0, z1))) → c27(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(sel(f(z0), x1)) → c27(SEL(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.42
PROPER(sel(cons(z0, z1), x1)) → c27(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(sel(g(z0), x1)) → c27(SEL(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.42
PROPER(sel(s(z0), x1)) → c27(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.42
PROPER(sel(sel(z0, z1), x1)) → c27(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(sel(x0, 0)) → c27(SEL(proper(x0), ok(0)), PROPER(x0)) 212.97/72.42
PROPER(sel(0, x1)) → c27(SEL(ok(0), proper(x1)), PROPER(x1)) 212.97/72.42
TOP(mark(f(z0))) → c28(TOP(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
TOP(mark(cons(z0, z1))) → c28(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
TOP(mark(g(z0))) → c28(TOP(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
TOP(mark(sel(z0, z1))) → c28(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.42
TOP(ok(f(z0))) → c29(TOP(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.42
TOP(ok(g(0))) → c29(TOP(mark(s(0))), ACTIVE(g(0))) 212.97/72.42
TOP(ok(g(s(z0)))) → c29(TOP(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.42
TOP(ok(sel(0, cons(z0, z1)))) → c29(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
TOP(ok(sel(s(z0), cons(z1, z2)))) → c29(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
TOP(ok(f(z0))) → c29(TOP(f(active(z0))), ACTIVE(f(z0))) 212.97/72.42
TOP(ok(cons(z0, z1))) → c29(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.42
TOP(ok(g(z0))) → c29(TOP(g(active(z0))), ACTIVE(g(z0))) 212.97/72.42
TOP(ok(s(z0))) → c29(TOP(s(active(z0))), ACTIVE(s(z0))) 212.97/72.42
TOP(ok(sel(z0, z1))) → c29(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.42
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, f, cons, g, s, sel, proper, top

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, c1, c2, c5, c6, c7, c3, c8, c25, c22, c22, c23, c23, c24, c24, c26, c26, c27, c27, c28, c28, c29

212.97/72.42

(67) 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)) 212.97/72.42
ACTIVE(sel(z0, z1)) → c9(SEL(active(z0), z1), ACTIVE(z0)) 212.97/72.42
ACTIVE(sel(z0, z1)) → c10(SEL(z0, active(z1)), ACTIVE(z1)) 212.97/72.42
ACTIVE(f(z0)) → c(G(z0)) 212.97/72.42
ACTIVE(g(0)) → c1 212.97/72.42
ACTIVE(g(s(z0))) → c2(G(z0)) 212.97/72.42
ACTIVE(f(f(z0))) → c5(F(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(f(g(s(z0)))) → c5(F(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(f(sel(0, cons(z0, z1)))) → c5(F(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(f(sel(s(z0), cons(z1, z2)))) → c5(F(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(f(f(z0))) → c5(F(f(active(z0))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(f(cons(z0, z1))) → c5(F(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(f(g(z0))) → c5(F(g(active(z0))), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(f(s(z0))) → c5(F(s(active(z0))), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(f(sel(z0, z1))) → c5(F(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(f(sel(z0, z1))) → c5(F(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(cons(f(z0), x1)) → c6(CONS(mark(cons(z0, f(g(z0)))), x1), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(cons(g(s(z0)), x1)) → c6(CONS(mark(s(s(g(z0)))), x1), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(cons(sel(0, cons(z0, z1)), x1)) → c6(CONS(mark(z0), x1), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(cons(sel(s(z0), cons(z1, z2)), x1)) → c6(CONS(mark(sel(z0, z2)), x1), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(cons(f(z0), x1)) → c6(CONS(f(active(z0)), x1), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(cons(cons(z0, z1), x1)) → c6(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(cons(g(z0), x1)) → c6(CONS(g(active(z0)), x1), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(cons(s(z0), x1)) → c6(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(active(z0), z1), x1), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(cons(sel(z0, z1), x1)) → c6(CONS(sel(z0, active(z1)), x1), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(g(f(z0))) → c7(G(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(g(g(s(z0)))) → c7(G(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(g(sel(0, cons(z0, z1)))) → c7(G(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(g(sel(s(z0), cons(z1, z2)))) → c7(G(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(g(f(z0))) → c7(G(f(active(z0))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(g(cons(z0, z1))) → c7(G(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(g(g(z0))) → c7(G(g(active(z0))), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(g(s(z0))) → c7(G(s(active(z0))), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(g(sel(z0, z1))) → c7(G(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(g(sel(z0, z1))) → c7(G(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(f(g(0))) → c3(F(mark(s(0)))) 212.97/72.42
ACTIVE(f(g(0))) → c3(ACTIVE(g(0))) 212.97/72.42
ACTIVE(cons(g(0), x1)) → c3(CONS(mark(s(0)), x1)) 212.97/72.42
ACTIVE(cons(g(0), x1)) → c3(ACTIVE(g(0))) 212.97/72.42
ACTIVE(g(g(0))) → c3(G(mark(s(0)))) 212.97/72.42
ACTIVE(g(g(0))) → c3(ACTIVE(g(0))) 212.97/72.42
ACTIVE(s(f(z0))) → c8(S(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(s(g(s(z0)))) → c8(S(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.42
ACTIVE(s(sel(0, cons(z0, z1)))) → c8(S(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
ACTIVE(s(sel(s(z0), cons(z1, z2)))) → c8(S(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
ACTIVE(s(f(z0))) → c8(S(f(active(z0))), ACTIVE(f(z0))) 212.97/72.42
ACTIVE(s(cons(z0, z1))) → c8(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.42
ACTIVE(s(g(z0))) → c8(S(g(active(z0))), ACTIVE(g(z0))) 212.97/72.42
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 212.97/72.42
ACTIVE(s(sel(z0, z1))) → c8(S(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(s(sel(z0, z1))) → c8(S(sel(z0, active(z1))), ACTIVE(sel(z0, z1))) 212.97/72.42
ACTIVE(s(g(0))) → c25(S(mark(s(0)))) 212.97/72.42
ACTIVE(s(g(0))) → c25(ACTIVE(g(0))) 212.97/72.42
PROPER(f(f(z0))) → c22(F(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
PROPER(f(cons(z0, z1))) → c22(F(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(f(g(z0))) → c22(F(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
PROPER(f(s(z0))) → c22(F(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
PROPER(f(sel(z0, z1))) → c22(F(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(f(0)) → c22(F(ok(0))) 212.97/72.42
PROPER(cons(x0, f(z0))) → c23(CONS(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.42
PROPER(cons(x0, cons(z0, z1))) → c23(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(cons(x0, g(z0))) → c23(CONS(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.42
PROPER(cons(x0, s(z0))) → c23(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.42
PROPER(cons(x0, sel(z0, z1))) → c23(CONS(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(cons(f(z0), x1)) → c23(CONS(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.42
PROPER(cons(cons(z0, z1), x1)) → c23(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(cons(g(z0), x1)) → c23(CONS(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.42
PROPER(cons(s(z0), x1)) → c23(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.42
PROPER(cons(sel(z0, z1), x1)) → c23(CONS(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(cons(x0, 0)) → c23(CONS(proper(x0), ok(0)), PROPER(x0)) 212.97/72.42
PROPER(cons(0, x1)) → c23(CONS(ok(0), proper(x1)), PROPER(x1)) 212.97/72.42
PROPER(g(f(z0))) → c24(G(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
PROPER(g(cons(z0, z1))) → c24(G(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(g(g(z0))) → c24(G(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
PROPER(g(s(z0))) → c24(G(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
PROPER(g(sel(z0, z1))) → c24(G(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(g(0)) → c24(G(ok(0))) 212.97/72.42
PROPER(s(f(z0))) → c26(S(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
PROPER(s(cons(z0, z1))) → c26(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(s(g(z0))) → c26(S(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
PROPER(s(s(z0))) → c26(S(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
PROPER(s(sel(z0, z1))) → c26(S(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(s(0)) → c26(S(ok(0))) 212.97/72.42
PROPER(sel(x0, f(z0))) → c27(SEL(proper(x0), f(proper(z0))), PROPER(x0), PROPER(f(z0))) 212.97/72.42
PROPER(sel(x0, cons(z0, z1))) → c27(SEL(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 212.97/72.42
PROPER(sel(x0, g(z0))) → c27(SEL(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0))) 212.97/72.42
PROPER(sel(x0, s(z0))) → c27(SEL(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 212.97/72.42
PROPER(sel(x0, sel(z0, z1))) → c27(SEL(proper(x0), sel(proper(z0), proper(z1))), PROPER(x0), PROPER(sel(z0, z1))) 212.97/72.42
PROPER(sel(f(z0), x1)) → c27(SEL(f(proper(z0)), proper(x1)), PROPER(f(z0)), PROPER(x1)) 212.97/72.42
PROPER(sel(cons(z0, z1), x1)) → c27(SEL(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(sel(g(z0), x1)) → c27(SEL(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1)) 212.97/72.42
PROPER(sel(s(z0), x1)) → c27(SEL(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 212.97/72.42
PROPER(sel(sel(z0, z1), x1)) → c27(SEL(sel(proper(z0), proper(z1)), proper(x1)), PROPER(sel(z0, z1)), PROPER(x1)) 212.97/72.42
PROPER(sel(x0, 0)) → c27(SEL(proper(x0), ok(0)), PROPER(x0)) 212.97/72.42
PROPER(sel(0, x1)) → c27(SEL(ok(0), proper(x1)), PROPER(x1)) 212.97/72.42
TOP(mark(f(z0))) → c28(TOP(f(proper(z0))), PROPER(f(z0))) 212.97/72.42
TOP(mark(cons(z0, z1))) → c28(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 212.97/72.42
TOP(mark(g(z0))) → c28(TOP(g(proper(z0))), PROPER(g(z0))) 212.97/72.42
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0))) 212.97/72.42
TOP(mark(sel(z0, z1))) → c28(TOP(sel(proper(z0), proper(z1))), PROPER(sel(z0, z1))) 212.97/72.42
TOP(ok(f(z0))) → c29(TOP(mark(cons(z0, f(g(z0))))), ACTIVE(f(z0))) 212.97/72.42
TOP(ok(g(0))) → c29(TOP(mark(s(0))), ACTIVE(g(0))) 212.97/72.42
TOP(ok(g(s(z0)))) → c29(TOP(mark(s(s(g(z0))))), ACTIVE(g(s(z0)))) 212.97/72.42
TOP(ok(sel(0, cons(z0, z1)))) → c29(TOP(mark(z0)), ACTIVE(sel(0, cons(z0, z1)))) 212.97/72.42
TOP(ok(sel(s(z0), cons(z1, z2)))) → c29(TOP(mark(sel(z0, z2))), ACTIVE(sel(s(z0), cons(z1, z2)))) 212.97/72.42
TOP(ok(f(z0))) → c29(TOP(f(active(z0))), ACTIVE(f(z0))) 212.97/72.42
TOP(ok(cons(z0, z1))) → c29(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 212.97/72.42
TOP(ok(g(z0))) → c29(TOP(g(active(z0))), ACTIVE(g(z0))) 212.97/72.42
TOP(ok(s(z0))) → c29(TOP(s(active(z0))), ACTIVE(s(z0))) 212.97/72.42
TOP(ok(sel(z0, z1))) → c29(TOP(sel(active(z0), z1)), ACTIVE(sel(z0, z1))) 212.97/72.42
TOP(ok(sel(z0, z1))) → c29(TOP(sel(z0, active(z1))), ACTIVE(sel(z0, z1)))

212.97/72.42

(68) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0)))) 212.97/72.42
active(g(0)) → mark(s(0)) 212.97/72.42
active(g(s(z0))) → mark(s(s(g(z0)))) 212.97/72.42
active(sel(0, cons(z0, z1))) → mark(z0) 212.97/72.42
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 212.97/72.42
active(f(z0)) → f(active(z0)) 212.97/72.42
active(cons(z0, z1)) → cons(active(z0), z1) 212.97/72.42
active(g(z0)) → g(active(z0)) 212.97/72.42
active(s(z0)) → s(active(z0)) 212.97/72.42
active(sel(z0, z1)) → sel(active(z0), z1) 212.97/72.42
active(sel(z0, z1)) → sel(z0, active(z1)) 212.97/72.42
f(mark(z0)) → mark(f(z0)) 212.97/72.42
f(ok(z0)) → ok(f(z0)) 212.97/72.42
cons(mark(z0), z1) → mark(cons(z0, z1)) 212.97/72.42
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 212.97/72.42
g(mark(z0)) → mark(g(z0)) 212.97/72.42
g(ok(z0)) → ok(g(z0)) 212.97/72.42
s(mark(z0)) → mark(s(z0)) 212.97/72.42
s(ok(z0)) → ok(s(z0)) 212.97/72.42
sel(mark(z0), z1) → mark(sel(z0, z1)) 212.97/72.42
sel(z0, mark(z1)) → mark(sel(z0, z1)) 212.97/72.42
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 212.97/72.42
proper(f(z0)) → f(proper(z0)) 212.97/72.42
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 212.97/72.42
proper(g(z0)) → g(proper(z0)) 212.97/72.42
proper(0) → ok(0) 212.97/72.42
proper(s(z0)) → s(proper(z0)) 212.97/72.42
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 212.97/72.42
top(mark(z0)) → top(proper(z0)) 212.97/72.42
top(ok(z0)) → top(active(z0))

Tuples:

F(mark(z0)) → c11(F(z0)) 212.97/72.42
F(ok(z0)) → c12(F(z0)) 212.97/72.42
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.42
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.43
G(mark(z0)) → c15(G(z0)) 212.97/72.43
G(ok(z0)) → c16(G(z0)) 212.97/72.43
S(mark(z0)) → c17(S(z0)) 212.97/72.43
S(ok(z0)) → c18(S(z0)) 212.97/72.43
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.43
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.43
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.97/72.43
TOP(mark(0)) → c28(TOP(ok(0)))

S tuples:

F(mark(z0)) → c11(F(z0)) 212.97/72.43
F(ok(z0)) → c12(F(z0)) 212.97/72.43
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.43
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.43
G(mark(z0)) → c15(G(z0)) 212.97/72.43
G(ok(z0)) → c16(G(z0)) 212.97/72.43
S(mark(z0)) → c17(S(z0)) 212.97/72.43
S(ok(z0)) → c18(S(z0)) 212.97/72.43
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.43
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.43
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))

K tuples:

TOP(mark(0)) → c28(TOP(ok(0)))

Defined Rule Symbols:

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

Defined Pair Symbols:

F, CONS, G, S, SEL, TOP

Compound Symbols:

c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c28

212.97/72.43

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

Removed 1 trailing tuple parts

212.97/72.43

(70) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0)))) 212.97/72.43
active(g(0)) → mark(s(0)) 212.97/72.43
active(g(s(z0))) → mark(s(s(g(z0)))) 212.97/72.43
active(sel(0, cons(z0, z1))) → mark(z0) 212.97/72.43
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 212.97/72.43
active(f(z0)) → f(active(z0)) 212.97/72.43
active(cons(z0, z1)) → cons(active(z0), z1) 212.97/72.43
active(g(z0)) → g(active(z0)) 212.97/72.43
active(s(z0)) → s(active(z0)) 212.97/72.43
active(sel(z0, z1)) → sel(active(z0), z1) 212.97/72.43
active(sel(z0, z1)) → sel(z0, active(z1)) 212.97/72.43
f(mark(z0)) → mark(f(z0)) 212.97/72.43
f(ok(z0)) → ok(f(z0)) 212.97/72.43
cons(mark(z0), z1) → mark(cons(z0, z1)) 212.97/72.43
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 212.97/72.43
g(mark(z0)) → mark(g(z0)) 212.97/72.43
g(ok(z0)) → ok(g(z0)) 212.97/72.43
s(mark(z0)) → mark(s(z0)) 212.97/72.43
s(ok(z0)) → ok(s(z0)) 212.97/72.43
sel(mark(z0), z1) → mark(sel(z0, z1)) 212.97/72.43
sel(z0, mark(z1)) → mark(sel(z0, z1)) 212.97/72.43
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 212.97/72.43
proper(f(z0)) → f(proper(z0)) 212.97/72.43
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 212.97/72.43
proper(g(z0)) → g(proper(z0)) 212.97/72.43
proper(0) → ok(0) 212.97/72.43
proper(s(z0)) → s(proper(z0)) 212.97/72.43
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 212.97/72.43
top(mark(z0)) → top(proper(z0)) 212.97/72.43
top(ok(z0)) → top(active(z0))

Tuples:

F(mark(z0)) → c11(F(z0)) 212.97/72.43
F(ok(z0)) → c12(F(z0)) 212.97/72.43
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.43
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.43
G(mark(z0)) → c15(G(z0)) 212.97/72.43
G(ok(z0)) → c16(G(z0)) 212.97/72.43
S(mark(z0)) → c17(S(z0)) 212.97/72.43
S(ok(z0)) → c18(S(z0)) 212.97/72.43
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.43
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.43
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.97/72.43
TOP(mark(0)) → c28

S tuples:

F(mark(z0)) → c11(F(z0)) 212.97/72.43
F(ok(z0)) → c12(F(z0)) 212.97/72.43
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.43
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.43
G(mark(z0)) → c15(G(z0)) 212.97/72.43
G(ok(z0)) → c16(G(z0)) 212.97/72.43
S(mark(z0)) → c17(S(z0)) 212.97/72.43
S(ok(z0)) → c18(S(z0)) 212.97/72.43
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.43
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.43
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))

K tuples:

TOP(mark(0)) → c28

Defined Rule Symbols:

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

Defined Pair Symbols:

F, CONS, G, S, SEL, TOP

Compound Symbols:

c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c28

212.97/72.43

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

Removed 1 trailing nodes:

TOP(mark(0)) → c28

212.97/72.43

(72) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0)))) 212.97/72.43
active(g(0)) → mark(s(0)) 212.97/72.43
active(g(s(z0))) → mark(s(s(g(z0)))) 212.97/72.43
active(sel(0, cons(z0, z1))) → mark(z0) 212.97/72.43
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 212.97/72.43
active(f(z0)) → f(active(z0)) 212.97/72.43
active(cons(z0, z1)) → cons(active(z0), z1) 212.97/72.43
active(g(z0)) → g(active(z0)) 212.97/72.43
active(s(z0)) → s(active(z0)) 212.97/72.43
active(sel(z0, z1)) → sel(active(z0), z1) 212.97/72.43
active(sel(z0, z1)) → sel(z0, active(z1)) 212.97/72.43
f(mark(z0)) → mark(f(z0)) 212.97/72.43
f(ok(z0)) → ok(f(z0)) 212.97/72.43
cons(mark(z0), z1) → mark(cons(z0, z1)) 212.97/72.43
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 212.97/72.43
g(mark(z0)) → mark(g(z0)) 212.97/72.43
g(ok(z0)) → ok(g(z0)) 212.97/72.43
s(mark(z0)) → mark(s(z0)) 212.97/72.43
s(ok(z0)) → ok(s(z0)) 212.97/72.43
sel(mark(z0), z1) → mark(sel(z0, z1)) 212.97/72.43
sel(z0, mark(z1)) → mark(sel(z0, z1)) 212.97/72.43
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 212.97/72.43
proper(f(z0)) → f(proper(z0)) 212.97/72.43
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 212.97/72.43
proper(g(z0)) → g(proper(z0)) 212.97/72.43
proper(0) → ok(0) 212.97/72.43
proper(s(z0)) → s(proper(z0)) 212.97/72.43
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 212.97/72.43
top(mark(z0)) → top(proper(z0)) 212.97/72.43
top(ok(z0)) → top(active(z0))

Tuples:

F(mark(z0)) → c11(F(z0)) 212.97/72.43
F(ok(z0)) → c12(F(z0)) 212.97/72.43
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.43
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.43
G(mark(z0)) → c15(G(z0)) 212.97/72.43
G(ok(z0)) → c16(G(z0)) 212.97/72.43
S(mark(z0)) → c17(S(z0)) 212.97/72.43
S(ok(z0)) → c18(S(z0)) 212.97/72.43
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.43
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.43
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))

S tuples:

F(mark(z0)) → c11(F(z0)) 212.97/72.43
F(ok(z0)) → c12(F(z0)) 212.97/72.43
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.43
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.43
G(mark(z0)) → c15(G(z0)) 212.97/72.43
G(ok(z0)) → c16(G(z0)) 212.97/72.43
S(mark(z0)) → c17(S(z0)) 212.97/72.43
S(ok(z0)) → c18(S(z0)) 212.97/72.43
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.43
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.43
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

F, CONS, G, S, SEL

Compound Symbols:

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

212.97/72.43

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

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

G(mark(z0)) → c15(G(z0))

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

F(mark(z0)) → c11(F(z0)) 212.97/72.43
F(ok(z0)) → c12(F(z0)) 212.97/72.43
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.43
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.43
G(mark(z0)) → c15(G(z0)) 212.97/72.43
G(ok(z0)) → c16(G(z0)) 212.97/72.43
S(mark(z0)) → c17(S(z0)) 212.97/72.43
S(ok(z0)) → c18(S(z0)) 212.97/72.43
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.43
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.43
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 212.97/72.43

POL(CONS(x₁, x₂)) = 0 212.97/72.43
POL(F(x₁)) = 0 212.97/72.43
POL(G(x₁)) = [4]x₁ 212.97/72.43
POL(S(x₁)) = 0 212.97/72.43
POL(SEL(x₁, x₂)) = 0 212.97/72.43
POL(c11(x₁)) = x₁ 212.97/72.43
POL(c12(x₁)) = x₁ 212.97/72.43
POL(c13(x₁)) = x₁ 212.97/72.43
POL(c14(x₁)) = x₁ 212.97/72.43
POL(c15(x₁)) = x₁ 212.97/72.43
POL(c16(x₁)) = x₁ 212.97/72.43
POL(c17(x₁)) = x₁ 212.97/72.43
POL(c18(x₁)) = x₁ 212.97/72.43
POL(c19(x₁)) = x₁ 212.97/72.43
POL(c20(x₁)) = x₁ 212.97/72.43
POL(c21(x₁)) = x₁ 212.97/72.43
POL(mark(x₁)) = [4] + x₁ 212.97/72.43
POL(ok(x₁)) = x₁

212.97/72.43

(74) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0)))) 212.97/72.43
active(g(0)) → mark(s(0)) 212.97/72.43
active(g(s(z0))) → mark(s(s(g(z0)))) 212.97/72.43
active(sel(0, cons(z0, z1))) → mark(z0) 212.97/72.43
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 212.97/72.43
active(f(z0)) → f(active(z0)) 212.97/72.43
active(cons(z0, z1)) → cons(active(z0), z1) 212.97/72.43
active(g(z0)) → g(active(z0)) 212.97/72.43
active(s(z0)) → s(active(z0)) 212.97/72.43
active(sel(z0, z1)) → sel(active(z0), z1) 212.97/72.43
active(sel(z0, z1)) → sel(z0, active(z1)) 212.97/72.43
f(mark(z0)) → mark(f(z0)) 212.97/72.43
f(ok(z0)) → ok(f(z0)) 212.97/72.43
cons(mark(z0), z1) → mark(cons(z0, z1)) 212.97/72.43
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 212.97/72.43
g(mark(z0)) → mark(g(z0)) 212.97/72.43
g(ok(z0)) → ok(g(z0)) 212.97/72.43
s(mark(z0)) → mark(s(z0)) 212.97/72.43
s(ok(z0)) → ok(s(z0)) 212.97/72.43
sel(mark(z0), z1) → mark(sel(z0, z1)) 212.97/72.43
sel(z0, mark(z1)) → mark(sel(z0, z1)) 212.97/72.43
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 212.97/72.43
proper(f(z0)) → f(proper(z0)) 212.97/72.43
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 212.97/72.43
proper(g(z0)) → g(proper(z0)) 212.97/72.43
proper(0) → ok(0) 212.97/72.43
proper(s(z0)) → s(proper(z0)) 212.97/72.43
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 212.97/72.43
top(mark(z0)) → top(proper(z0)) 212.97/72.43
top(ok(z0)) → top(active(z0))

Tuples:

F(mark(z0)) → c11(F(z0)) 212.97/72.43
F(ok(z0)) → c12(F(z0)) 212.97/72.43
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.43
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.43
G(mark(z0)) → c15(G(z0)) 212.97/72.43
G(ok(z0)) → c16(G(z0)) 212.97/72.43
S(mark(z0)) → c17(S(z0)) 212.97/72.43
S(ok(z0)) → c18(S(z0)) 212.97/72.43
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.43
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.43
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))

S tuples:

F(mark(z0)) → c11(F(z0)) 212.97/72.43
F(ok(z0)) → c12(F(z0)) 212.97/72.43
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.43
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.43
G(ok(z0)) → c16(G(z0)) 212.97/72.43
S(mark(z0)) → c17(S(z0)) 212.97/72.43
S(ok(z0)) → c18(S(z0)) 212.97/72.43
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.43
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.43
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))

K tuples:

G(mark(z0)) → c15(G(z0))

Defined Rule Symbols:

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

Defined Pair Symbols:

F, CONS, G, S, SEL

Compound Symbols:

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

212.97/72.43

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

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

S(ok(z0)) → c18(S(z0))

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

F(mark(z0)) → c11(F(z0)) 212.97/72.43
F(ok(z0)) → c12(F(z0)) 212.97/72.43
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.43
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.43
G(mark(z0)) → c15(G(z0)) 212.97/72.43
G(ok(z0)) → c16(G(z0)) 212.97/72.43
S(mark(z0)) → c17(S(z0)) 212.97/72.43
S(ok(z0)) → c18(S(z0)) 212.97/72.43
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.43
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.43
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 212.97/72.43

POL(CONS(x₁, x₂)) = 0 212.97/72.43
POL(F(x₁)) = 0 212.97/72.43
POL(G(x₁)) = 0 212.97/72.43
POL(S(x₁)) = [2]x₁ 212.97/72.43
POL(SEL(x₁, x₂)) = 0 212.97/72.43
POL(c11(x₁)) = x₁ 212.97/72.43
POL(c12(x₁)) = x₁ 212.97/72.43
POL(c13(x₁)) = x₁ 212.97/72.43
POL(c14(x₁)) = x₁ 212.97/72.43
POL(c15(x₁)) = x₁ 212.97/72.43
POL(c16(x₁)) = x₁ 212.97/72.43
POL(c17(x₁)) = x₁ 212.97/72.43
POL(c18(x₁)) = x₁ 212.97/72.43
POL(c19(x₁)) = x₁ 212.97/72.43
POL(c20(x₁)) = x₁ 212.97/72.43
POL(c21(x₁)) = x₁ 212.97/72.43
POL(mark(x₁)) = x₁ 212.97/72.43
POL(ok(x₁)) = [1] + x₁

212.97/72.43

(76) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0)))) 212.97/72.43
active(g(0)) → mark(s(0)) 212.97/72.43
active(g(s(z0))) → mark(s(s(g(z0)))) 212.97/72.43
active(sel(0, cons(z0, z1))) → mark(z0) 212.97/72.43
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 212.97/72.43
active(f(z0)) → f(active(z0)) 212.97/72.43
active(cons(z0, z1)) → cons(active(z0), z1) 212.97/72.43
active(g(z0)) → g(active(z0)) 212.97/72.43
active(s(z0)) → s(active(z0)) 212.97/72.43
active(sel(z0, z1)) → sel(active(z0), z1) 212.97/72.43
active(sel(z0, z1)) → sel(z0, active(z1)) 212.97/72.43
f(mark(z0)) → mark(f(z0)) 212.97/72.43
f(ok(z0)) → ok(f(z0)) 212.97/72.43
cons(mark(z0), z1) → mark(cons(z0, z1)) 212.97/72.43
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 212.97/72.43
g(mark(z0)) → mark(g(z0)) 212.97/72.43
g(ok(z0)) → ok(g(z0)) 212.97/72.43
s(mark(z0)) → mark(s(z0)) 212.97/72.43
s(ok(z0)) → ok(s(z0)) 212.97/72.43
sel(mark(z0), z1) → mark(sel(z0, z1)) 212.97/72.43
sel(z0, mark(z1)) → mark(sel(z0, z1)) 212.97/72.43
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 212.97/72.43
proper(f(z0)) → f(proper(z0)) 212.97/72.43
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 212.97/72.43
proper(g(z0)) → g(proper(z0)) 212.97/72.43
proper(0) → ok(0) 212.97/72.43
proper(s(z0)) → s(proper(z0)) 212.97/72.43
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 212.97/72.43
top(mark(z0)) → top(proper(z0)) 212.97/72.43
top(ok(z0)) → top(active(z0))

Tuples:

F(mark(z0)) → c11(F(z0)) 212.97/72.43
F(ok(z0)) → c12(F(z0)) 212.97/72.43
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.43
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.43
G(mark(z0)) → c15(G(z0)) 212.97/72.43
G(ok(z0)) → c16(G(z0)) 212.97/72.43
S(mark(z0)) → c17(S(z0)) 212.97/72.43
S(ok(z0)) → c18(S(z0)) 212.97/72.43
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.43
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.43
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))

S tuples:

F(mark(z0)) → c11(F(z0)) 212.97/72.43
F(ok(z0)) → c12(F(z0)) 212.97/72.43
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.43
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.43
G(ok(z0)) → c16(G(z0)) 212.97/72.43
S(mark(z0)) → c17(S(z0)) 212.97/72.43
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.43
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.43
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))

K tuples:

G(mark(z0)) → c15(G(z0)) 212.97/72.43
S(ok(z0)) → c18(S(z0))

Defined Rule Symbols:

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

Defined Pair Symbols:

F, CONS, G, S, SEL

Compound Symbols:

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

212.97/72.43

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

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

G(ok(z0)) → c16(G(z0)) 212.97/72.43
S(mark(z0)) → c17(S(z0)) 212.97/72.43
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.43
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.43
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))

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

F(mark(z0)) → c11(F(z0)) 212.97/72.43
F(ok(z0)) → c12(F(z0)) 212.97/72.43
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.43
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.43
G(mark(z0)) → c15(G(z0)) 212.97/72.43
G(ok(z0)) → c16(G(z0)) 212.97/72.43
S(mark(z0)) → c17(S(z0)) 212.97/72.43
S(ok(z0)) → c18(S(z0)) 212.97/72.43
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.43
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.43
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 212.97/72.43

POL(CONS(x₁, x₂)) = 0 212.97/72.43
POL(F(x₁)) = 0 212.97/72.43
POL(G(x₁)) = x₁ 212.97/72.43
POL(S(x₁)) = x₁ 212.97/72.43
POL(SEL(x₁, x₂)) = x₁ + [2]x₂ 212.97/72.43
POL(c11(x₁)) = x₁ 212.97/72.43
POL(c12(x₁)) = x₁ 212.97/72.43
POL(c13(x₁)) = x₁ 212.97/72.43
POL(c14(x₁)) = x₁ 212.97/72.43
POL(c15(x₁)) = x₁ 212.97/72.43
POL(c16(x₁)) = x₁ 212.97/72.43
POL(c17(x₁)) = x₁ 212.97/72.43
POL(c18(x₁)) = x₁ 212.97/72.43
POL(c19(x₁)) = x₁ 212.97/72.43
POL(c20(x₁)) = x₁ 212.97/72.43
POL(c21(x₁)) = x₁ 212.97/72.43
POL(mark(x₁)) = [4] + x₁ 212.97/72.43
POL(ok(x₁)) = [1] + x₁

212.97/72.43

(78) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0)))) 212.97/72.43
active(g(0)) → mark(s(0)) 212.97/72.43
active(g(s(z0))) → mark(s(s(g(z0)))) 212.97/72.43
active(sel(0, cons(z0, z1))) → mark(z0) 212.97/72.43
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 212.97/72.43
active(f(z0)) → f(active(z0)) 212.97/72.43
active(cons(z0, z1)) → cons(active(z0), z1) 212.97/72.43
active(g(z0)) → g(active(z0)) 212.97/72.43
active(s(z0)) → s(active(z0)) 212.97/72.43
active(sel(z0, z1)) → sel(active(z0), z1) 212.97/72.43
active(sel(z0, z1)) → sel(z0, active(z1)) 212.97/72.43
f(mark(z0)) → mark(f(z0)) 212.97/72.43
f(ok(z0)) → ok(f(z0)) 212.97/72.43
cons(mark(z0), z1) → mark(cons(z0, z1)) 212.97/72.43
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 212.97/72.43
g(mark(z0)) → mark(g(z0)) 212.97/72.43
g(ok(z0)) → ok(g(z0)) 212.97/72.43
s(mark(z0)) → mark(s(z0)) 212.97/72.43
s(ok(z0)) → ok(s(z0)) 212.97/72.43
sel(mark(z0), z1) → mark(sel(z0, z1)) 212.97/72.43
sel(z0, mark(z1)) → mark(sel(z0, z1)) 212.97/72.43
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 212.97/72.43
proper(f(z0)) → f(proper(z0)) 212.97/72.43
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 212.97/72.43
proper(g(z0)) → g(proper(z0)) 212.97/72.43
proper(0) → ok(0) 212.97/72.43
proper(s(z0)) → s(proper(z0)) 212.97/72.43
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 212.97/72.43
top(mark(z0)) → top(proper(z0)) 212.97/72.43
top(ok(z0)) → top(active(z0))

Tuples:

F(mark(z0)) → c11(F(z0)) 212.97/72.43
F(ok(z0)) → c12(F(z0)) 212.97/72.43
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.43
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.43
G(mark(z0)) → c15(G(z0)) 212.97/72.43
G(ok(z0)) → c16(G(z0)) 212.97/72.43
S(mark(z0)) → c17(S(z0)) 212.97/72.43
S(ok(z0)) → c18(S(z0)) 212.97/72.43
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.43
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.43
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))

S tuples:

F(mark(z0)) → c11(F(z0)) 212.97/72.43
F(ok(z0)) → c12(F(z0)) 212.97/72.43
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.43
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))

K tuples:

G(mark(z0)) → c15(G(z0)) 212.97/72.43
S(ok(z0)) → c18(S(z0)) 212.97/72.43
G(ok(z0)) → c16(G(z0)) 212.97/72.43
S(mark(z0)) → c17(S(z0)) 212.97/72.43
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.43
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.43
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))

Defined Rule Symbols:

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

Defined Pair Symbols:

F, CONS, G, S, SEL

Compound Symbols:

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

212.97/72.43

(79) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^3))) transformation)

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

F(mark(z0)) → c11(F(z0)) 212.97/72.43
F(ok(z0)) → c12(F(z0)) 212.97/72.43
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.43
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))

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

F(mark(z0)) → c11(F(z0)) 212.97/72.43
F(ok(z0)) → c12(F(z0)) 212.97/72.43
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.43
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.43
G(mark(z0)) → c15(G(z0)) 212.97/72.43
G(ok(z0)) → c16(G(z0)) 212.97/72.43
S(mark(z0)) → c17(S(z0)) 212.97/72.43
S(ok(z0)) → c18(S(z0)) 212.97/72.43
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.43
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.43
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 212.97/72.43

POL(CONS(x₁, x₂)) = x₁ + x₂² + x₁·x₂ + x₁² + x₁²·x₂ + x₂³ 212.97/72.43
POL(F(x₁)) = x₁ + x₁² + x₁³ 212.97/72.43
POL(G(x₁)) = x₁ + x₁² + x₁³ 212.97/72.43
POL(S(x₁)) = x₁ + x₁² + x₁³ 212.97/72.43
POL(SEL(x₁, x₂)) = x₁ + x₂ + x₂² + x₁·x₂ + x₁² + x₁³ + x₁²·x₂ + x₁·x₂² + x₂³ 212.97/72.43
POL(c11(x₁)) = x₁ 212.97/72.43
POL(c12(x₁)) = x₁ 212.97/72.43
POL(c13(x₁)) = x₁ 212.97/72.43
POL(c14(x₁)) = x₁ 212.97/72.43
POL(c15(x₁)) = x₁ 212.97/72.43
POL(c16(x₁)) = x₁ 212.97/72.43
POL(c17(x₁)) = x₁ 212.97/72.43
POL(c18(x₁)) = x₁ 212.97/72.43
POL(c19(x₁)) = x₁ 212.97/72.43
POL(c20(x₁)) = x₁ 212.97/72.43
POL(c21(x₁)) = x₁ 212.97/72.43
POL(mark(x₁)) = [1] + x₁ 212.97/72.43
POL(ok(x₁)) = [1] + x₁

212.97/72.43

(80) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(cons(z0, f(g(z0)))) 212.97/72.43
active(g(0)) → mark(s(0)) 212.97/72.43
active(g(s(z0))) → mark(s(s(g(z0)))) 212.97/72.43
active(sel(0, cons(z0, z1))) → mark(z0) 212.97/72.43
active(sel(s(z0), cons(z1, z2))) → mark(sel(z0, z2)) 212.97/72.43
active(f(z0)) → f(active(z0)) 212.97/72.43
active(cons(z0, z1)) → cons(active(z0), z1) 212.97/72.43
active(g(z0)) → g(active(z0)) 212.97/72.43
active(s(z0)) → s(active(z0)) 212.97/72.43
active(sel(z0, z1)) → sel(active(z0), z1) 212.97/72.43
active(sel(z0, z1)) → sel(z0, active(z1)) 212.97/72.43
f(mark(z0)) → mark(f(z0)) 212.97/72.43
f(ok(z0)) → ok(f(z0)) 212.97/72.43
cons(mark(z0), z1) → mark(cons(z0, z1)) 212.97/72.43
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 212.97/72.43
g(mark(z0)) → mark(g(z0)) 212.97/72.43
g(ok(z0)) → ok(g(z0)) 212.97/72.43
s(mark(z0)) → mark(s(z0)) 212.97/72.43
s(ok(z0)) → ok(s(z0)) 212.97/72.43
sel(mark(z0), z1) → mark(sel(z0, z1)) 212.97/72.43
sel(z0, mark(z1)) → mark(sel(z0, z1)) 212.97/72.43
sel(ok(z0), ok(z1)) → ok(sel(z0, z1)) 212.97/72.43
proper(f(z0)) → f(proper(z0)) 212.97/72.43
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 212.97/72.43
proper(g(z0)) → g(proper(z0)) 212.97/72.43
proper(0) → ok(0) 212.97/72.43
proper(s(z0)) → s(proper(z0)) 212.97/72.43
proper(sel(z0, z1)) → sel(proper(z0), proper(z1)) 212.97/72.43
top(mark(z0)) → top(proper(z0)) 212.97/72.43
top(ok(z0)) → top(active(z0))

Tuples:

F(mark(z0)) → c11(F(z0)) 212.97/72.43
F(ok(z0)) → c12(F(z0)) 212.97/72.43
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.43
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1)) 212.97/72.43
G(mark(z0)) → c15(G(z0)) 212.97/72.43
G(ok(z0)) → c16(G(z0)) 212.97/72.43
S(mark(z0)) → c17(S(z0)) 212.97/72.43
S(ok(z0)) → c18(S(z0)) 212.97/72.43
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.43
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.43
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1))

S tuples:none
K tuples:

G(mark(z0)) → c15(G(z0)) 212.97/72.43
S(ok(z0)) → c18(S(z0)) 212.97/72.43
G(ok(z0)) → c16(G(z0)) 212.97/72.43
S(mark(z0)) → c17(S(z0)) 212.97/72.43
SEL(mark(z0), z1) → c19(SEL(z0, z1)) 212.97/72.43
SEL(z0, mark(z1)) → c20(SEL(z0, z1)) 212.97/72.43
SEL(ok(z0), ok(z1)) → c21(SEL(z0, z1)) 212.97/72.43
F(mark(z0)) → c11(F(z0)) 212.97/72.43
F(ok(z0)) → c12(F(z0)) 212.97/72.43
CONS(mark(z0), z1) → c13(CONS(z0, z1)) 212.97/72.43
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))

Defined Rule Symbols:

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

Defined Pair Symbols:

F, CONS, G, S, SEL

Compound Symbols:

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

212.97/72.43

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

The set S is empty

212.97/72.43

(82) BOUNDS(O(1), O(1))

212.97/72.43