Runtime Complexity (innermost) proof of /scratch/tmpsUoAdj/Ex14_Luc06

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

0 CpxTRS

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

↳2 CdtProblem

↳3 CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication)

↳4 CdtProblem

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

↳6 CdtProblem

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

↳8 CdtProblem

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

↳10 CdtProblem

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

↳12 CdtProblem

↳13 CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication)

↳14 CdtProblem

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

↳16 CdtProblem

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

↳18 CdtProblem

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

↳20 CdtProblem

↳21 CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication)

↳22 CdtProblem

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

↳24 CdtProblem

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

↳26 CdtProblem

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

↳28 CdtProblem

↳29 CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication)

↳30 CdtProblem

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

↳32 CdtProblem

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

↳34 CdtProblem

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

↳36 CdtProblem

↳37 CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication)

↳38 CdtProblem

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

↳40 CdtProblem

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

↳42 CdtProblem

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

↳44 CdtProblem

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

↳46 CdtProblem

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

↳48 CdtProblem

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

↳50 CdtProblem

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

↳52 CdtProblem

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

↳54 CdtProblem

↳55 CdtUnreachableProof (⇔)

↳56 CdtProblem

↳57 CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication)

↳58 CdtProblem

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

↳60 CdtProblem

↳61 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))

↳62 CdtProblem

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

↳64 CdtProblem

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

↳66 CdtProblem

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

↳68 CdtProblem

↳69 SIsEmptyProof (⇔)

↳70 BOUNDS(O(1), O(1))

(0) Obligation:

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

active(h(X)) → mark(g(X, X))
active(g(a, X)) → mark(f(b, X))
active(f(X, X)) → mark(h(a))
active(a) → mark(b)
active(h(X)) → h(active(X))
active(g(X1, X2)) → g(active(X1), X2)
active(f(X1, X2)) → f(active(X1), X2)
h(mark(X)) → mark(h(X))
g(mark(X1), X2) → mark(g(X1, X2))
f(mark(X1), X2) → mark(f(X1, X2))
proper(h(X)) → h(proper(X))
proper(g(X1, X2)) → g(proper(X1), proper(X2))
proper(a) → ok(a)
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(b) → ok(b)
h(ok(X)) → ok(h(X))
g(ok(X1), ok(X2)) → ok(g(X1, X2))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0))
active(g(a, z0)) → mark(f(b, z0))
active(f(z0, z0)) → mark(h(a))
active(a) → mark(b)
active(h(z0)) → h(active(z0))
active(g(z0, z1)) → g(active(z0), z1)
active(f(z0, z1)) → f(active(z0), z1)
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
g(mark(z0), z1) → mark(g(z0, z1))
g(ok(z0), ok(z1)) → ok(g(z0, z1))
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(h(z0)) → h(proper(z0))
proper(g(z0, z1)) → g(proper(z0), proper(z1))
proper(a) → ok(a)
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
ACTIVE(g(a, z0)) → c1(F(b, z0))
ACTIVE(f(z0, z0)) → c2(H(a))
ACTIVE(h(z0)) → c4(H(active(z0)), ACTIVE(z0))
ACTIVE(g(z0, z1)) → c5(G(active(z0), z1), ACTIVE(z0))
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
ACTIVE(g(a, z0)) → c1(F(b, z0))
ACTIVE(f(z0, z0)) → c2(H(a))
ACTIVE(h(z0)) → c4(H(active(z0)), ACTIVE(z0))
ACTIVE(g(z0, z1)) → c5(G(active(z0), z1), ACTIVE(z0))
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c1, c2, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19

(3) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 2 of 17 dangling nodes:

ACTIVE(g(a, z0)) → c1(F(b, z0))
ACTIVE(f(z0, z0)) → c2(H(a))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0))
active(g(a, z0)) → mark(f(b, z0))
active(f(z0, z0)) → mark(h(a))
active(a) → mark(b)
active(h(z0)) → h(active(z0))
active(g(z0, z1)) → g(active(z0), z1)
active(f(z0, z1)) → f(active(z0), z1)
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
g(mark(z0), z1) → mark(g(z0, z1))
g(ok(z0), ok(z1)) → ok(g(z0, z1))
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(h(z0)) → h(proper(z0))
proper(g(z0, z1)) → g(proper(z0), proper(z1))
proper(a) → ok(a)
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
ACTIVE(h(z0)) → c4(H(active(z0)), ACTIVE(z0))
ACTIVE(g(z0, z1)) → c5(G(active(z0), z1), ACTIVE(z0))
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
ACTIVE(h(z0)) → c4(H(active(z0)), ACTIVE(z0))
ACTIVE(g(z0, z1)) → c5(G(active(z0), z1), ACTIVE(z0))
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19

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

Use narrowing to replace ACTIVE(h(z0)) → c4(H(active(z0)), ACTIVE(z0)) by

ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(g(a, z0))) → c4(H(mark(f(b, z0))), ACTIVE(g(a, z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(a)) → c4(H(mark(b)), ACTIVE(a))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0))
active(g(a, z0)) → mark(f(b, z0))
active(f(z0, z0)) → mark(h(a))
active(a) → mark(b)
active(h(z0)) → h(active(z0))
active(g(z0, z1)) → g(active(z0), z1)
active(f(z0, z1)) → f(active(z0), z1)
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
g(mark(z0), z1) → mark(g(z0, z1))
g(ok(z0), ok(z1)) → ok(g(z0, z1))
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(h(z0)) → h(proper(z0))
proper(g(z0, z1)) → g(proper(z0), proper(z1))
proper(a) → ok(a)
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
ACTIVE(g(z0, z1)) → c5(G(active(z0), z1), ACTIVE(z0))
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(g(a, z0))) → c4(H(mark(f(b, z0))), ACTIVE(g(a, z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(a)) → c4(H(mark(b)), ACTIVE(a))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))

S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
ACTIVE(g(z0, z1)) → c5(G(active(z0), z1), ACTIVE(z0))
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(g(a, z0))) → c4(H(mark(f(b, z0))), ACTIVE(g(a, z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(a)) → c4(H(mark(b)), ACTIVE(a))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))

K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c4

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

Split RHS of tuples not part of any SCC

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0))
active(g(a, z0)) → mark(f(b, z0))
active(f(z0, z0)) → mark(h(a))
active(a) → mark(b)
active(h(z0)) → h(active(z0))
active(g(z0, z1)) → g(active(z0), z1)
active(f(z0, z1)) → f(active(z0), z1)
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
g(mark(z0), z1) → mark(g(z0, z1))
g(ok(z0), ok(z1)) → ok(g(z0, z1))
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(h(z0)) → h(proper(z0))
proper(g(z0, z1)) → g(proper(z0), proper(z1))
proper(a) → ok(a)
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
ACTIVE(g(z0, z1)) → c5(G(active(z0), z1), ACTIVE(z0))
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(g(a, z0))) → c4(H(mark(f(b, z0))), ACTIVE(g(a, z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(h(a)) → c1(ACTIVE(a))

S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
ACTIVE(g(z0, z1)) → c5(G(active(z0), z1), ACTIVE(z0))
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(g(a, z0))) → c4(H(mark(f(b, z0))), ACTIVE(g(a, z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(h(a)) → c1(ACTIVE(a))

K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c4, c1

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

Removed 1 trailing tuple parts

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0))
active(g(a, z0)) → mark(f(b, z0))
active(f(z0, z0)) → mark(h(a))
active(a) → mark(b)
active(h(z0)) → h(active(z0))
active(g(z0, z1)) → g(active(z0), z1)
active(f(z0, z1)) → f(active(z0), z1)
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
g(mark(z0), z1) → mark(g(z0, z1))
g(ok(z0), ok(z1)) → ok(g(z0, z1))
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(h(z0)) → h(proper(z0))
proper(g(z0, z1)) → g(proper(z0), proper(z1))
proper(a) → ok(a)
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
ACTIVE(g(z0, z1)) → c5(G(active(z0), z1), ACTIVE(z0))
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(g(a, z0))) → c4(H(mark(f(b, z0))), ACTIVE(g(a, z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(h(a)) → c1

S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
ACTIVE(g(z0, z1)) → c5(G(active(z0), z1), ACTIVE(z0))
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(g(a, z0))) → c4(H(mark(f(b, z0))), ACTIVE(g(a, z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(h(a)) → c1

K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c4, c1, c1

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

Use narrowing to replace ACTIVE(g(z0, z1)) → c5(G(active(z0), z1), ACTIVE(z0)) by

ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(a, z0), x1)) → c5(G(mark(f(b, z0)), x1), ACTIVE(g(a, z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(a, x1)) → c5(G(mark(b), x1), ACTIVE(a))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0))
active(g(a, z0)) → mark(f(b, z0))
active(f(z0, z0)) → mark(h(a))
active(a) → mark(b)
active(h(z0)) → h(active(z0))
active(g(z0, z1)) → g(active(z0), z1)
active(f(z0, z1)) → f(active(z0), z1)
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
g(mark(z0), z1) → mark(g(z0, z1))
g(ok(z0), ok(z1)) → ok(g(z0, z1))
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(h(z0)) → h(proper(z0))
proper(g(z0, z1)) → g(proper(z0), proper(z1))
proper(a) → ok(a)
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(g(a, z0))) → c4(H(mark(f(b, z0))), ACTIVE(g(a, z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(h(a)) → c1
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(a, z0), x1)) → c5(G(mark(f(b, z0)), x1), ACTIVE(g(a, z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(a, x1)) → c5(G(mark(b), x1), ACTIVE(a))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))

S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(g(a, z0))) → c4(H(mark(f(b, z0))), ACTIVE(g(a, z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(h(a)) → c1
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(a, z0), x1)) → c5(G(mark(f(b, z0)), x1), ACTIVE(g(a, z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(a, x1)) → c5(G(mark(b), x1), ACTIVE(a))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))

K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c6, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c4, c1, c1, c5

(13) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 28 dangling nodes:

ACTIVE(h(a)) → c1

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0))
active(g(a, z0)) → mark(f(b, z0))
active(f(z0, z0)) → mark(h(a))
active(a) → mark(b)
active(h(z0)) → h(active(z0))
active(g(z0, z1)) → g(active(z0), z1)
active(f(z0, z1)) → f(active(z0), z1)
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
g(mark(z0), z1) → mark(g(z0, z1))
g(ok(z0), ok(z1)) → ok(g(z0, z1))
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(h(z0)) → h(proper(z0))
proper(g(z0, z1)) → g(proper(z0), proper(z1))
proper(a) → ok(a)
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(g(a, z0))) → c4(H(mark(f(b, z0))), ACTIVE(g(a, z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(a, z0), x1)) → c5(G(mark(f(b, z0)), x1), ACTIVE(g(a, z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(a, x1)) → c5(G(mark(b), x1), ACTIVE(a))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))

S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(g(a, z0))) → c4(H(mark(f(b, z0))), ACTIVE(g(a, z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(a, z0), x1)) → c5(G(mark(f(b, z0)), x1), ACTIVE(g(a, z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(a, x1)) → c5(G(mark(b), x1), ACTIVE(a))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))

K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c6, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c4, c1, c5

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

Split RHS of tuples not part of any SCC

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0))
active(g(a, z0)) → mark(f(b, z0))
active(f(z0, z0)) → mark(h(a))
active(a) → mark(b)
active(h(z0)) → h(active(z0))
active(g(z0, z1)) → g(active(z0), z1)
active(f(z0, z1)) → f(active(z0), z1)
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
g(mark(z0), z1) → mark(g(z0, z1))
g(ok(z0), ok(z1)) → ok(g(z0, z1))
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(h(z0)) → h(proper(z0))
proper(g(z0, z1)) → g(proper(z0), proper(z1))
proper(a) → ok(a)
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(h(g(a, z0))) → c2(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c2(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(a, x1)) → c2(G(mark(b), x1))
ACTIVE(g(a, x1)) → c2(ACTIVE(a))

S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(h(g(a, z0))) → c2(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c2(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(a, x1)) → c2(G(mark(b), x1))
ACTIVE(g(a, x1)) → c2(ACTIVE(a))

K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c6, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c4, c1, c5, c2

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

Removed 1 trailing tuple parts

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0))
active(g(a, z0)) → mark(f(b, z0))
active(f(z0, z0)) → mark(h(a))
active(a) → mark(b)
active(h(z0)) → h(active(z0))
active(g(z0, z1)) → g(active(z0), z1)
active(f(z0, z1)) → f(active(z0), z1)
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
g(mark(z0), z1) → mark(g(z0, z1))
g(ok(z0), ok(z1)) → ok(g(z0, z1))
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(h(z0)) → h(proper(z0))
proper(g(z0, z1)) → g(proper(z0), proper(z1))
proper(a) → ok(a)
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(h(g(a, z0))) → c2(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c2(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(a, x1)) → c2(G(mark(b), x1))
ACTIVE(g(a, x1)) → c2

S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(h(g(a, z0))) → c2(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c2(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(a, x1)) → c2(G(mark(b), x1))
ACTIVE(g(a, x1)) → c2

K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c6, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c4, c1, c5, c2, c2

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

Use narrowing to replace ACTIVE(f(z0, z1)) → c6(F(active(z0), z1), ACTIVE(z0)) by

ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(a, z0), x1)) → c6(F(mark(f(b, z0)), x1), ACTIVE(g(a, z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(a, x1)) → c6(F(mark(b), x1), ACTIVE(a))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0))
active(g(a, z0)) → mark(f(b, z0))
active(f(z0, z0)) → mark(h(a))
active(a) → mark(b)
active(h(z0)) → h(active(z0))
active(g(z0, z1)) → g(active(z0), z1)
active(f(z0, z1)) → f(active(z0), z1)
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
g(mark(z0), z1) → mark(g(z0, z1))
g(ok(z0), ok(z1)) → ok(g(z0, z1))
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(h(z0)) → h(proper(z0))
proper(g(z0, z1)) → g(proper(z0), proper(z1))
proper(a) → ok(a)
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(h(g(a, z0))) → c2(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c2(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(a, x1)) → c2(G(mark(b), x1))
ACTIVE(g(a, x1)) → c2
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(a, z0), x1)) → c6(F(mark(f(b, z0)), x1), ACTIVE(g(a, z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(a, x1)) → c6(F(mark(b), x1), ACTIVE(a))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))

S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(h(g(a, z0))) → c2(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c2(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(a, x1)) → c2(G(mark(b), x1))
ACTIVE(g(a, x1)) → c2
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(a, z0), x1)) → c6(F(mark(f(b, z0)), x1), ACTIVE(g(a, z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(a, x1)) → c6(F(mark(b), x1), ACTIVE(a))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))

K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c4, c1, c5, c2, c2, c6

(21) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 36 dangling nodes:

ACTIVE(g(a, x1)) → c2

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0))
active(g(a, z0)) → mark(f(b, z0))
active(f(z0, z0)) → mark(h(a))
active(a) → mark(b)
active(h(z0)) → h(active(z0))
active(g(z0, z1)) → g(active(z0), z1)
active(f(z0, z1)) → f(active(z0), z1)
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
g(mark(z0), z1) → mark(g(z0, z1))
g(ok(z0), ok(z1)) → ok(g(z0, z1))
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(h(z0)) → h(proper(z0))
proper(g(z0, z1)) → g(proper(z0), proper(z1))
proper(a) → ok(a)
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(h(g(a, z0))) → c2(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c2(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(a, x1)) → c2(G(mark(b), x1))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(a, z0), x1)) → c6(F(mark(f(b, z0)), x1), ACTIVE(g(a, z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(a, x1)) → c6(F(mark(b), x1), ACTIVE(a))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))

S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(h(g(a, z0))) → c2(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c2(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(a, x1)) → c2(G(mark(b), x1))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(a, z0), x1)) → c6(F(mark(f(b, z0)), x1), ACTIVE(g(a, z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(a, x1)) → c6(F(mark(b), x1), ACTIVE(a))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))

K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c4, c1, c5, c2, c6

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

Split RHS of tuples not part of any SCC

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0))
active(g(a, z0)) → mark(f(b, z0))
active(f(z0, z0)) → mark(h(a))
active(a) → mark(b)
active(h(z0)) → h(active(z0))
active(g(z0, z1)) → g(active(z0), z1)
active(f(z0, z1)) → f(active(z0), z1)
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
g(mark(z0), z1) → mark(g(z0, z1))
g(ok(z0), ok(z1)) → ok(g(z0, z1))
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(h(z0)) → h(proper(z0))
proper(g(z0, z1)) → g(proper(z0), proper(z1))
proper(a) → ok(a)
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(h(g(a, z0))) → c2(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c2(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(a, x1)) → c2(G(mark(b), x1))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(a, z0), x1)) → c3(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c3(ACTIVE(g(a, z0)))
ACTIVE(f(a, x1)) → c3(F(mark(b), x1))
ACTIVE(f(a, x1)) → c3(ACTIVE(a))

S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(h(g(a, z0))) → c2(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c2(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(a, x1)) → c2(G(mark(b), x1))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(a, z0), x1)) → c3(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c3(ACTIVE(g(a, z0)))
ACTIVE(f(a, x1)) → c3(F(mark(b), x1))
ACTIVE(f(a, x1)) → c3(ACTIVE(a))

K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c4, c1, c5, c2, c6, c3

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

Removed 1 trailing tuple parts

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0))
active(g(a, z0)) → mark(f(b, z0))
active(f(z0, z0)) → mark(h(a))
active(a) → mark(b)
active(h(z0)) → h(active(z0))
active(g(z0, z1)) → g(active(z0), z1)
active(f(z0, z1)) → f(active(z0), z1)
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
g(mark(z0), z1) → mark(g(z0, z1))
g(ok(z0), ok(z1)) → ok(g(z0, z1))
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(h(z0)) → h(proper(z0))
proper(g(z0, z1)) → g(proper(z0), proper(z1))
proper(a) → ok(a)
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(h(g(a, z0))) → c2(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c2(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(a, x1)) → c2(G(mark(b), x1))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(a, z0), x1)) → c3(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c3(ACTIVE(g(a, z0)))
ACTIVE(f(a, x1)) → c3(F(mark(b), x1))
ACTIVE(f(a, x1)) → c3

S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(h(g(a, z0))) → c2(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c2(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(a, x1)) → c2(G(mark(b), x1))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(a, z0), x1)) → c3(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c3(ACTIVE(g(a, z0)))
ACTIVE(f(a, x1)) → c3(F(mark(b), x1))
ACTIVE(f(a, x1)) → c3

K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c7, c8, c9, c10, c11, c12, c13, c14, c16, c18, c19, c4, c1, c5, c2, c6, c3, c3

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

Use narrowing to replace PROPER(h(z0)) → c13(H(proper(z0)), PROPER(z0)) by

PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(a)) → c13(H(ok(a)), PROPER(a))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(b)) → c13(H(ok(b)), PROPER(b))

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0))
active(g(a, z0)) → mark(f(b, z0))
active(f(z0, z0)) → mark(h(a))
active(a) → mark(b)
active(h(z0)) → h(active(z0))
active(g(z0, z1)) → g(active(z0), z1)
active(f(z0, z1)) → f(active(z0), z1)
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
g(mark(z0), z1) → mark(g(z0, z1))
g(ok(z0), ok(z1)) → ok(g(z0, z1))
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(h(z0)) → h(proper(z0))
proper(g(z0, z1)) → g(proper(z0), proper(z1))
proper(a) → ok(a)
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(h(g(a, z0))) → c2(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c2(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(a, x1)) → c2(G(mark(b), x1))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(a, z0), x1)) → c3(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c3(ACTIVE(g(a, z0)))
ACTIVE(f(a, x1)) → c3(F(mark(b), x1))
ACTIVE(f(a, x1)) → c3
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(a)) → c13(H(ok(a)), PROPER(a))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(b)) → c13(H(ok(b)), PROPER(b))

S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(h(g(a, z0))) → c2(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c2(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(a, x1)) → c2(G(mark(b), x1))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(a, z0), x1)) → c3(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c3(ACTIVE(g(a, z0)))
ACTIVE(f(a, x1)) → c3(F(mark(b), x1))
ACTIVE(f(a, x1)) → c3
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(a)) → c13(H(ok(a)), PROPER(a))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(b)) → c13(H(ok(b)), PROPER(b))

K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c7, c8, c9, c10, c11, c12, c14, c16, c18, c19, c4, c1, c5, c2, c6, c3, c3, c13

(29) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 41 dangling nodes:

ACTIVE(f(a, x1)) → c3

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0))
active(g(a, z0)) → mark(f(b, z0))
active(f(z0, z0)) → mark(h(a))
active(a) → mark(b)
active(h(z0)) → h(active(z0))
active(g(z0, z1)) → g(active(z0), z1)
active(f(z0, z1)) → f(active(z0), z1)
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
g(mark(z0), z1) → mark(g(z0, z1))
g(ok(z0), ok(z1)) → ok(g(z0, z1))
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(h(z0)) → h(proper(z0))
proper(g(z0, z1)) → g(proper(z0), proper(z1))
proper(a) → ok(a)
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(h(g(a, z0))) → c2(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c2(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(a, x1)) → c2(G(mark(b), x1))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(a, z0), x1)) → c3(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c3(ACTIVE(g(a, z0)))
ACTIVE(f(a, x1)) → c3(F(mark(b), x1))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(a)) → c13(H(ok(a)), PROPER(a))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(b)) → c13(H(ok(b)), PROPER(b))

S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(h(g(a, z0))) → c2(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c2(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(a, x1)) → c2(G(mark(b), x1))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(a, z0), x1)) → c3(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c3(ACTIVE(g(a, z0)))
ACTIVE(f(a, x1)) → c3(F(mark(b), x1))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(a)) → c13(H(ok(a)), PROPER(a))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(b)) → c13(H(ok(b)), PROPER(b))

K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

c, c7, c8, c9, c10, c11, c12, c14, c16, c18, c19, c4, c1, c5, c2, c6, c3, c13

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

Split RHS of tuples not part of any SCC

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0))
active(g(a, z0)) → mark(f(b, z0))
active(f(z0, z0)) → mark(h(a))
active(a) → mark(b)
active(h(z0)) → h(active(z0))
active(g(z0, z1)) → g(active(z0), z1)
active(f(z0, z1)) → f(active(z0), z1)
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
g(mark(z0), z1) → mark(g(z0, z1))
g(ok(z0), ok(z1)) → ok(g(z0, z1))
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(h(z0)) → h(proper(z0))
proper(g(z0, z1)) → g(proper(z0), proper(z1))
proper(a) → ok(a)
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(h(g(a, z0))) → c2(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c2(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(a, x1)) → c2(G(mark(b), x1))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(a, z0), x1)) → c3(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c3(ACTIVE(g(a, z0)))
ACTIVE(f(a, x1)) → c3(F(mark(b), x1))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c15(H(ok(a)))
PROPER(h(a)) → c15(PROPER(a))
PROPER(h(b)) → c15(H(ok(b)))
PROPER(h(b)) → c15(PROPER(b))

S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(h(g(a, z0))) → c2(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c2(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(a, x1)) → c2(G(mark(b), x1))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(a, z0), x1)) → c3(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c3(ACTIVE(g(a, z0)))
ACTIVE(f(a, x1)) → c3(F(mark(b), x1))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c15(H(ok(a)))
PROPER(h(a)) → c15(PROPER(a))
PROPER(h(b)) → c15(H(ok(b)))
PROPER(h(b)) → c15(PROPER(b))

K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

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

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

Removed 2 trailing tuple parts

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0))
active(g(a, z0)) → mark(f(b, z0))
active(f(z0, z0)) → mark(h(a))
active(a) → mark(b)
active(h(z0)) → h(active(z0))
active(g(z0, z1)) → g(active(z0), z1)
active(f(z0, z1)) → f(active(z0), z1)
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
g(mark(z0), z1) → mark(g(z0, z1))
g(ok(z0), ok(z1)) → ok(g(z0, z1))
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(h(z0)) → h(proper(z0))
proper(g(z0, z1)) → g(proper(z0), proper(z1))
proper(a) → ok(a)
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(h(g(a, z0))) → c2(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c2(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(a, x1)) → c2(G(mark(b), x1))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(a, z0), x1)) → c3(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c3(ACTIVE(g(a, z0)))
ACTIVE(f(a, x1)) → c3(F(mark(b), x1))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c15(H(ok(a)))
PROPER(h(b)) → c15(H(ok(b)))
PROPER(h(a)) → c15
PROPER(h(b)) → c15

S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(h(g(a, z0))) → c2(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c2(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(a, x1)) → c2(G(mark(b), x1))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(a, z0), x1)) → c3(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c3(ACTIVE(g(a, z0)))
ACTIVE(f(a, x1)) → c3(F(mark(b), x1))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c15(H(ok(a)))
PROPER(h(b)) → c15(H(ok(b)))
PROPER(h(a)) → c15
PROPER(h(b)) → c15

K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

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

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

Use narrowing to replace PROPER(g(z0, z1)) → c14(G(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0), PROPER(a))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0), PROPER(b))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(a), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(b), PROPER(x1))

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0))
active(g(a, z0)) → mark(f(b, z0))
active(f(z0, z0)) → mark(h(a))
active(a) → mark(b)
active(h(z0)) → h(active(z0))
active(g(z0, z1)) → g(active(z0), z1)
active(f(z0, z1)) → f(active(z0), z1)
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
g(mark(z0), z1) → mark(g(z0, z1))
g(ok(z0), ok(z1)) → ok(g(z0, z1))
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(h(z0)) → h(proper(z0))
proper(g(z0, z1)) → g(proper(z0), proper(z1))
proper(a) → ok(a)
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(h(g(a, z0))) → c2(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c2(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(a, x1)) → c2(G(mark(b), x1))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(a, z0), x1)) → c3(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c3(ACTIVE(g(a, z0)))
ACTIVE(f(a, x1)) → c3(F(mark(b), x1))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c15(H(ok(a)))
PROPER(h(b)) → c15(H(ok(b)))
PROPER(h(a)) → c15
PROPER(h(b)) → c15
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0), PROPER(a))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0), PROPER(b))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(a), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(b), PROPER(x1))

S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(h(g(a, z0))) → c2(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c2(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(a, x1)) → c2(G(mark(b), x1))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(a, z0), x1)) → c3(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c3(ACTIVE(g(a, z0)))
ACTIVE(f(a, x1)) → c3(F(mark(b), x1))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c15(H(ok(a)))
PROPER(h(b)) → c15(H(ok(b)))
PROPER(h(a)) → c15
PROPER(h(b)) → c15
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0), PROPER(a))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0), PROPER(b))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(a), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(b), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

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

(37) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 2 of 51 dangling nodes:

PROPER(h(b)) → c15
PROPER(h(a)) → c15

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0))
active(g(a, z0)) → mark(f(b, z0))
active(f(z0, z0)) → mark(h(a))
active(a) → mark(b)
active(h(z0)) → h(active(z0))
active(g(z0, z1)) → g(active(z0), z1)
active(f(z0, z1)) → f(active(z0), z1)
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
g(mark(z0), z1) → mark(g(z0, z1))
g(ok(z0), ok(z1)) → ok(g(z0, z1))
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(h(z0)) → h(proper(z0))
proper(g(z0, z1)) → g(proper(z0), proper(z1))
proper(a) → ok(a)
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(h(g(a, z0))) → c2(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c2(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(a, x1)) → c2(G(mark(b), x1))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(a, z0), x1)) → c3(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c3(ACTIVE(g(a, z0)))
ACTIVE(f(a, x1)) → c3(F(mark(b), x1))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c15(H(ok(a)))
PROPER(h(b)) → c15(H(ok(b)))
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0), PROPER(a))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0), PROPER(b))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(a), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(b), PROPER(x1))

S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(h(g(a, z0))) → c2(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c2(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(a, x1)) → c2(G(mark(b), x1))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(a, z0), x1)) → c3(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c3(ACTIVE(g(a, z0)))
ACTIVE(f(a, x1)) → c3(F(mark(b), x1))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c15(H(ok(a)))
PROPER(h(b)) → c15(H(ok(b)))
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0), PROPER(a))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0), PROPER(b))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(a), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(b), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

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

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

Removed 4 trailing tuple parts

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0))
active(g(a, z0)) → mark(f(b, z0))
active(f(z0, z0)) → mark(h(a))
active(a) → mark(b)
active(h(z0)) → h(active(z0))
active(g(z0, z1)) → g(active(z0), z1)
active(f(z0, z1)) → f(active(z0), z1)
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
g(mark(z0), z1) → mark(g(z0, z1))
g(ok(z0), ok(z1)) → ok(g(z0, z1))
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(h(z0)) → h(proper(z0))
proper(g(z0, z1)) → g(proper(z0), proper(z1))
proper(a) → ok(a)
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(h(g(a, z0))) → c2(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c2(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(a, x1)) → c2(G(mark(b), x1))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(a, z0), x1)) → c3(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c3(ACTIVE(g(a, z0)))
ACTIVE(f(a, x1)) → c3(F(mark(b), x1))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c15(H(ok(a)))
PROPER(h(b)) → c15(H(ok(b)))
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1))

S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(h(g(a, z0))) → c2(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c2(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(a, x1)) → c2(G(mark(b), x1))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(a, z0), x1)) → c3(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c3(ACTIVE(g(a, z0)))
ACTIVE(f(a, x1)) → c3(F(mark(b), x1))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c15(H(ok(a)))
PROPER(h(b)) → c15(H(ok(b)))
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

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

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

Use narrowing to replace PROPER(f(z0, z1)) → c16(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0), PROPER(a))
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0), PROPER(b))
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(a), PROPER(x1))
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(b), PROPER(x1))

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0))
active(g(a, z0)) → mark(f(b, z0))
active(f(z0, z0)) → mark(h(a))
active(a) → mark(b)
active(h(z0)) → h(active(z0))
active(g(z0, z1)) → g(active(z0), z1)
active(f(z0, z1)) → f(active(z0), z1)
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
g(mark(z0), z1) → mark(g(z0, z1))
g(ok(z0), ok(z1)) → ok(g(z0, z1))
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(h(z0)) → h(proper(z0))
proper(g(z0, z1)) → g(proper(z0), proper(z1))
proper(a) → ok(a)
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(h(g(a, z0))) → c2(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c2(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(a, x1)) → c2(G(mark(b), x1))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(a, z0), x1)) → c3(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c3(ACTIVE(g(a, z0)))
ACTIVE(f(a, x1)) → c3(F(mark(b), x1))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c15(H(ok(a)))
PROPER(h(b)) → c15(H(ok(b)))
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1))
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0), PROPER(a))
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0), PROPER(b))
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(a), PROPER(x1))
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(b), PROPER(x1))

S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(h(g(a, z0))) → c2(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c2(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(a, x1)) → c2(G(mark(b), x1))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(a, z0), x1)) → c3(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c3(ACTIVE(g(a, z0)))
ACTIVE(f(a, x1)) → c3(F(mark(b), x1))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c15(H(ok(a)))
PROPER(h(b)) → c15(H(ok(b)))
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1))
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0), PROPER(a))
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0), PROPER(b))
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(a), PROPER(x1))
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(b), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, TOP, PROPER

Compound Symbols:

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

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

Removed 4 trailing tuple parts

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0))
active(g(a, z0)) → mark(f(b, z0))
active(f(z0, z0)) → mark(h(a))
active(a) → mark(b)
active(h(z0)) → h(active(z0))
active(g(z0, z1)) → g(active(z0), z1)
active(f(z0, z1)) → f(active(z0), z1)
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
g(mark(z0), z1) → mark(g(z0, z1))
g(ok(z0), ok(z1)) → ok(g(z0, z1))
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(h(z0)) → h(proper(z0))
proper(g(z0, z1)) → g(proper(z0), proper(z1))
proper(a) → ok(a)
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(h(g(a, z0))) → c2(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c2(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(a, x1)) → c2(G(mark(b), x1))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(a, z0), x1)) → c3(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c3(ACTIVE(g(a, z0)))
ACTIVE(f(a, x1)) → c3(F(mark(b), x1))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c15(H(ok(a)))
PROPER(h(b)) → c15(H(ok(b)))
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1))
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1))

S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(h(g(a, z0))) → c2(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c2(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(a, x1)) → c2(G(mark(b), x1))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(a, z0), x1)) → c3(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c3(ACTIVE(g(a, z0)))
ACTIVE(f(a, x1)) → c3(F(mark(b), x1))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c15(H(ok(a)))
PROPER(h(b)) → c15(H(ok(b)))
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1))
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, TOP, PROPER

Compound Symbols:

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

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

Use narrowing to replace TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(h(z0))) → c18(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(g(z0, z1))) → c18(TOP(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
TOP(mark(a)) → c18(TOP(ok(a)), PROPER(a))
TOP(mark(f(z0, z1))) → c18(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(b)) → c18(TOP(ok(b)), PROPER(b))

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0))
active(g(a, z0)) → mark(f(b, z0))
active(f(z0, z0)) → mark(h(a))
active(a) → mark(b)
active(h(z0)) → h(active(z0))
active(g(z0, z1)) → g(active(z0), z1)
active(f(z0, z1)) → f(active(z0), z1)
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
g(mark(z0), z1) → mark(g(z0, z1))
g(ok(z0), ok(z1)) → ok(g(z0, z1))
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(h(z0)) → h(proper(z0))
proper(g(z0, z1)) → g(proper(z0), proper(z1))
proper(a) → ok(a)
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(h(g(a, z0))) → c2(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c2(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(a, x1)) → c2(G(mark(b), x1))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(a, z0), x1)) → c3(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c3(ACTIVE(g(a, z0)))
ACTIVE(f(a, x1)) → c3(F(mark(b), x1))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c15(H(ok(a)))
PROPER(h(b)) → c15(H(ok(b)))
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1))
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(h(z0))) → c18(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(g(z0, z1))) → c18(TOP(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
TOP(mark(a)) → c18(TOP(ok(a)), PROPER(a))
TOP(mark(f(z0, z1))) → c18(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(b)) → c18(TOP(ok(b)), PROPER(b))

S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(h(g(a, z0))) → c2(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c2(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(a, x1)) → c2(G(mark(b), x1))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(a, z0), x1)) → c3(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c3(ACTIVE(g(a, z0)))
ACTIVE(f(a, x1)) → c3(F(mark(b), x1))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c15(H(ok(a)))
PROPER(h(b)) → c15(H(ok(b)))
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1))
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(h(z0))) → c18(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(g(z0, z1))) → c18(TOP(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
TOP(mark(a)) → c18(TOP(ok(a)), PROPER(a))
TOP(mark(f(z0, z1))) → c18(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(b)) → c18(TOP(ok(b)), PROPER(b))

K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, TOP, PROPER

Compound Symbols:

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

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

Removed 2 trailing tuple parts

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0))
active(g(a, z0)) → mark(f(b, z0))
active(f(z0, z0)) → mark(h(a))
active(a) → mark(b)
active(h(z0)) → h(active(z0))
active(g(z0, z1)) → g(active(z0), z1)
active(f(z0, z1)) → f(active(z0), z1)
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
g(mark(z0), z1) → mark(g(z0, z1))
g(ok(z0), ok(z1)) → ok(g(z0, z1))
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(h(z0)) → h(proper(z0))
proper(g(z0, z1)) → g(proper(z0), proper(z1))
proper(a) → ok(a)
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(h(g(a, z0))) → c2(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c2(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(a, x1)) → c2(G(mark(b), x1))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(a, z0), x1)) → c3(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c3(ACTIVE(g(a, z0)))
ACTIVE(f(a, x1)) → c3(F(mark(b), x1))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c15(H(ok(a)))
PROPER(h(b)) → c15(H(ok(b)))
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1))
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(h(z0))) → c18(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(g(z0, z1))) → c18(TOP(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
TOP(mark(f(z0, z1))) → c18(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(a)) → c18(TOP(ok(a)))
TOP(mark(b)) → c18(TOP(ok(b)))

S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(h(g(a, z0))) → c2(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c2(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(a, x1)) → c2(G(mark(b), x1))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(a, z0), x1)) → c3(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c3(ACTIVE(g(a, z0)))
ACTIVE(f(a, x1)) → c3(F(mark(b), x1))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c15(H(ok(a)))
PROPER(h(b)) → c15(H(ok(b)))
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1))
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(h(z0))) → c18(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(g(z0, z1))) → c18(TOP(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
TOP(mark(f(z0, z1))) → c18(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(a)) → c18(TOP(ok(a)))
TOP(mark(b)) → c18(TOP(ok(b)))

K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, TOP, PROPER

Compound Symbols:

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

(49) 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(a)) → c18(TOP(ok(a)))

We considered the (Usable) Rules:

proper(h(z0)) → h(proper(z0))
proper(g(z0, z1)) → g(proper(z0), proper(z1))
proper(a) → ok(a)
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(b) → ok(b)
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
g(mark(z0), z1) → mark(g(z0, z1))
g(ok(z0), ok(z1)) → ok(g(z0, z1))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
active(h(z0)) → mark(g(z0, z0))
active(g(a, z0)) → mark(f(b, z0))
active(f(z0, z0)) → mark(h(a))
active(a) → mark(b)
active(h(z0)) → h(active(z0))
active(g(z0, z1)) → g(active(z0), z1)
active(f(z0, z1)) → f(active(z0), z1)

And the Tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(h(g(a, z0))) → c2(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c2(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(a, x1)) → c2(G(mark(b), x1))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(a, z0), x1)) → c3(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c3(ACTIVE(g(a, z0)))
ACTIVE(f(a, x1)) → c3(F(mark(b), x1))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c15(H(ok(a)))
PROPER(h(b)) → c15(H(ok(b)))
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1))
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(h(z0))) → c18(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(g(z0, z1))) → c18(TOP(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
TOP(mark(f(z0, z1))) → c18(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(a)) → c18(TOP(ok(a)))
TOP(mark(b)) → c18(TOP(ok(b)))

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

POL(ACTIVE(x₁)) = 0
POL(F(x₁, x₂)) = 0
POL(G(x₁, x₂)) = 0
POL(H(x₁)) = 0
POL(PROPER(x₁)) = 0
POL(TOP(x₁)) = x₁
POL(a) = [4]
POL(active(x₁)) = 0
POL(b) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁, x₂)) = x₁ + x₂
POL(c14(x₁, x₂)) = x₁ + x₂
POL(c14(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c15(x₁)) = x₁
POL(c16(x₁, x₂)) = x₁ + x₂
POL(c16(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c18(x₁)) = x₁
POL(c18(x₁, x₂)) = x₁ + x₂
POL(c19(x₁, x₂)) = x₁ + x₂
POL(c2(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c4(x₁, x₂)) = x₁ + x₂
POL(c5(x₁, x₂)) = x₁ + x₂
POL(c6(x₁, x₂)) = x₁ + x₂
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(f(x₁, x₂)) = 0
POL(g(x₁, x₂)) = 0
POL(h(x₁)) = 0
POL(mark(x₁)) = x₁
POL(ok(x₁)) = 0
POL(proper(x₁)) = 0

(50) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0))
active(g(a, z0)) → mark(f(b, z0))
active(f(z0, z0)) → mark(h(a))
active(a) → mark(b)
active(h(z0)) → h(active(z0))
active(g(z0, z1)) → g(active(z0), z1)
active(f(z0, z1)) → f(active(z0), z1)
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
g(mark(z0), z1) → mark(g(z0, z1))
g(ok(z0), ok(z1)) → ok(g(z0, z1))
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(h(z0)) → h(proper(z0))
proper(g(z0, z1)) → g(proper(z0), proper(z1))
proper(a) → ok(a)
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(h(g(a, z0))) → c2(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c2(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(a, x1)) → c2(G(mark(b), x1))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(a, z0), x1)) → c3(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c3(ACTIVE(g(a, z0)))
ACTIVE(f(a, x1)) → c3(F(mark(b), x1))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c15(H(ok(a)))
PROPER(h(b)) → c15(H(ok(b)))
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1))
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(h(z0))) → c18(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(g(z0, z1))) → c18(TOP(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
TOP(mark(f(z0, z1))) → c18(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(a)) → c18(TOP(ok(a)))
TOP(mark(b)) → c18(TOP(ok(b)))

S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(h(g(a, z0))) → c2(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c2(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(a, x1)) → c2(G(mark(b), x1))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(a, z0), x1)) → c3(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c3(ACTIVE(g(a, z0)))
ACTIVE(f(a, x1)) → c3(F(mark(b), x1))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c15(H(ok(a)))
PROPER(h(b)) → c15(H(ok(b)))
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1))
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(h(z0))) → c18(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(g(z0, z1))) → c18(TOP(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
TOP(mark(f(z0, z1))) → c18(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(b)) → c18(TOP(ok(b)))

K tuples:

TOP(mark(a)) → c18(TOP(ok(a)))

Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, TOP, PROPER

Compound Symbols:

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

(51) 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(b)) → c18(TOP(ok(b)))

We considered the (Usable) Rules:

proper(h(z0)) → h(proper(z0))
proper(g(z0, z1)) → g(proper(z0), proper(z1))
proper(a) → ok(a)
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(b) → ok(b)
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
g(mark(z0), z1) → mark(g(z0, z1))
g(ok(z0), ok(z1)) → ok(g(z0, z1))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
active(h(z0)) → mark(g(z0, z0))
active(g(a, z0)) → mark(f(b, z0))
active(f(z0, z0)) → mark(h(a))
active(a) → mark(b)
active(h(z0)) → h(active(z0))
active(g(z0, z1)) → g(active(z0), z1)
active(f(z0, z1)) → f(active(z0), z1)

And the Tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(h(g(a, z0))) → c2(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c2(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(a, x1)) → c2(G(mark(b), x1))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(a, z0), x1)) → c3(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c3(ACTIVE(g(a, z0)))
ACTIVE(f(a, x1)) → c3(F(mark(b), x1))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c15(H(ok(a)))
PROPER(h(b)) → c15(H(ok(b)))
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1))
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(h(z0))) → c18(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(g(z0, z1))) → c18(TOP(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
TOP(mark(f(z0, z1))) → c18(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(a)) → c18(TOP(ok(a)))
TOP(mark(b)) → c18(TOP(ok(b)))

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

POL(ACTIVE(x₁)) = 0
POL(F(x₁, x₂)) = 0
POL(G(x₁, x₂)) = 0
POL(H(x₁)) = 0
POL(PROPER(x₁)) = 0
POL(TOP(x₁)) = x₁
POL(a) = [1]
POL(active(x₁)) = x₁
POL(b) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁, x₂)) = x₁ + x₂
POL(c14(x₁, x₂)) = x₁ + x₂
POL(c14(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c15(x₁)) = x₁
POL(c16(x₁, x₂)) = x₁ + x₂
POL(c16(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c18(x₁)) = x₁
POL(c18(x₁, x₂)) = x₁ + x₂
POL(c19(x₁, x₂)) = x₁ + x₂
POL(c2(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c4(x₁, x₂)) = x₁ + x₂
POL(c5(x₁, x₂)) = x₁ + x₂
POL(c6(x₁, x₂)) = x₁ + x₂
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(f(x₁, x₂)) = [1]
POL(g(x₁, x₂)) = [1]
POL(h(x₁)) = [1]
POL(mark(x₁)) = [1]
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0

(52) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0))
active(g(a, z0)) → mark(f(b, z0))
active(f(z0, z0)) → mark(h(a))
active(a) → mark(b)
active(h(z0)) → h(active(z0))
active(g(z0, z1)) → g(active(z0), z1)
active(f(z0, z1)) → f(active(z0), z1)
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
g(mark(z0), z1) → mark(g(z0, z1))
g(ok(z0), ok(z1)) → ok(g(z0, z1))
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(h(z0)) → h(proper(z0))
proper(g(z0, z1)) → g(proper(z0), proper(z1))
proper(a) → ok(a)
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(h(g(a, z0))) → c2(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c2(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(a, x1)) → c2(G(mark(b), x1))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(a, z0), x1)) → c3(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c3(ACTIVE(g(a, z0)))
ACTIVE(f(a, x1)) → c3(F(mark(b), x1))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c15(H(ok(a)))
PROPER(h(b)) → c15(H(ok(b)))
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1))
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(h(z0))) → c18(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(g(z0, z1))) → c18(TOP(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
TOP(mark(f(z0, z1))) → c18(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(a)) → c18(TOP(ok(a)))
TOP(mark(b)) → c18(TOP(ok(b)))

S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(h(g(a, z0))) → c2(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c2(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(a, x1)) → c2(G(mark(b), x1))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(a, z0), x1)) → c3(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c3(ACTIVE(g(a, z0)))
ACTIVE(f(a, x1)) → c3(F(mark(b), x1))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c15(H(ok(a)))
PROPER(h(b)) → c15(H(ok(b)))
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1))
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(h(z0))) → c18(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(g(z0, z1))) → c18(TOP(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
TOP(mark(f(z0, z1))) → c18(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))

K tuples:

TOP(mark(a)) → c18(TOP(ok(a)))
TOP(mark(b)) → c18(TOP(ok(b)))

Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, TOP, PROPER

Compound Symbols:

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

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

Use narrowing to replace TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(h(z0))) → c19(TOP(mark(g(z0, z0))), ACTIVE(h(z0)))
TOP(ok(g(a, z0))) → c19(TOP(mark(f(b, z0))), ACTIVE(g(a, z0)))
TOP(ok(f(z0, z0))) → c19(TOP(mark(h(a))), ACTIVE(f(z0, z0)))
TOP(ok(a)) → c19(TOP(mark(b)), ACTIVE(a))
TOP(ok(h(z0))) → c19(TOP(h(active(z0))), ACTIVE(h(z0)))
TOP(ok(g(z0, z1))) → c19(TOP(g(active(z0), z1)), ACTIVE(g(z0, z1)))
TOP(ok(f(z0, z1))) → c19(TOP(f(active(z0), z1)), ACTIVE(f(z0, z1)))

(54) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0))
active(g(a, z0)) → mark(f(b, z0))
active(f(z0, z0)) → mark(h(a))
active(a) → mark(b)
active(h(z0)) → h(active(z0))
active(g(z0, z1)) → g(active(z0), z1)
active(f(z0, z1)) → f(active(z0), z1)
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
g(mark(z0), z1) → mark(g(z0, z1))
g(ok(z0), ok(z1)) → ok(g(z0, z1))
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(h(z0)) → h(proper(z0))
proper(g(z0, z1)) → g(proper(z0), proper(z1))
proper(a) → ok(a)
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(h(g(a, z0))) → c2(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c2(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(a, x1)) → c2(G(mark(b), x1))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(a, z0), x1)) → c3(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c3(ACTIVE(g(a, z0)))
ACTIVE(f(a, x1)) → c3(F(mark(b), x1))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c15(H(ok(a)))
PROPER(h(b)) → c15(H(ok(b)))
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1))
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(h(z0))) → c18(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(g(z0, z1))) → c18(TOP(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
TOP(mark(f(z0, z1))) → c18(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(a)) → c18(TOP(ok(a)))
TOP(mark(b)) → c18(TOP(ok(b)))
TOP(ok(h(z0))) → c19(TOP(mark(g(z0, z0))), ACTIVE(h(z0)))
TOP(ok(g(a, z0))) → c19(TOP(mark(f(b, z0))), ACTIVE(g(a, z0)))
TOP(ok(f(z0, z0))) → c19(TOP(mark(h(a))), ACTIVE(f(z0, z0)))
TOP(ok(a)) → c19(TOP(mark(b)), ACTIVE(a))
TOP(ok(h(z0))) → c19(TOP(h(active(z0))), ACTIVE(h(z0)))
TOP(ok(g(z0, z1))) → c19(TOP(g(active(z0), z1)), ACTIVE(g(z0, z1)))
TOP(ok(f(z0, z1))) → c19(TOP(f(active(z0), z1)), ACTIVE(f(z0, z1)))

S tuples:

ACTIVE(h(z0)) → c(G(z0, z0))
H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(h(g(a, z0))) → c2(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c2(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(a, x1)) → c2(G(mark(b), x1))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(a, z0), x1)) → c3(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c3(ACTIVE(g(a, z0)))
ACTIVE(f(a, x1)) → c3(F(mark(b), x1))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c15(H(ok(a)))
PROPER(h(b)) → c15(H(ok(b)))
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1))
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(h(z0))) → c18(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(g(z0, z1))) → c18(TOP(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
TOP(mark(f(z0, z1))) → c18(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(ok(h(z0))) → c19(TOP(mark(g(z0, z0))), ACTIVE(h(z0)))
TOP(ok(g(a, z0))) → c19(TOP(mark(f(b, z0))), ACTIVE(g(a, z0)))
TOP(ok(f(z0, z0))) → c19(TOP(mark(h(a))), ACTIVE(f(z0, z0)))
TOP(ok(a)) → c19(TOP(mark(b)), ACTIVE(a))
TOP(ok(h(z0))) → c19(TOP(h(active(z0))), ACTIVE(h(z0)))
TOP(ok(g(z0, z1))) → c19(TOP(g(active(z0), z1)), ACTIVE(g(z0, z1)))
TOP(ok(f(z0, z1))) → c19(TOP(f(active(z0), z1)), ACTIVE(f(z0, z1)))

K tuples:

TOP(mark(a)) → c18(TOP(ok(a)))
TOP(mark(b)) → c18(TOP(ok(b)))

Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

ACTIVE, H, G, F, PROPER, TOP

Compound Symbols:

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

(55) CdtUnreachableProof (EQUIVALENT transformation)

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

ACTIVE(h(z0)) → c(G(z0, z0))
ACTIVE(h(h(z0))) → c4(H(mark(g(z0, z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0, z0))) → c4(H(mark(h(a))), ACTIVE(f(z0, z0)))
ACTIVE(h(h(z0))) → c4(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(g(z0, z1))) → c4(H(g(active(z0), z1)), ACTIVE(g(z0, z1)))
ACTIVE(h(f(z0, z1))) → c4(H(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(h(a)) → c1(H(mark(b)))
ACTIVE(g(h(z0), x1)) → c5(G(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(f(z0, z0), x1)) → c5(G(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(g(h(z0), x1)) → c5(G(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(g(g(z0, z1), x1)) → c5(G(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(g(f(z0, z1), x1)) → c5(G(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(h(g(a, z0))) → c2(H(mark(f(b, z0))))
ACTIVE(h(g(a, z0))) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(g(a, z0), x1)) → c2(G(mark(f(b, z0)), x1))
ACTIVE(g(g(a, z0), x1)) → c2(ACTIVE(g(a, z0)))
ACTIVE(g(a, x1)) → c2(G(mark(b), x1))
ACTIVE(f(h(z0), x1)) → c6(F(mark(g(z0, z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(f(z0, z0), x1)) → c6(F(mark(h(a)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(h(z0), x1)) → c6(F(h(active(z0)), x1), ACTIVE(h(z0)))
ACTIVE(f(g(z0, z1), x1)) → c6(F(g(active(z0), z1), x1), ACTIVE(g(z0, z1)))
ACTIVE(f(f(z0, z1), x1)) → c6(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(a, z0), x1)) → c3(F(mark(f(b, z0)), x1))
ACTIVE(f(g(a, z0), x1)) → c3(ACTIVE(g(a, z0)))
ACTIVE(f(a, x1)) → c3(F(mark(b), x1))
PROPER(h(h(z0))) → c13(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(g(z0, z1))) → c13(H(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
PROPER(h(f(z0, z1))) → c13(H(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(h(a)) → c15(H(ok(a)))
PROPER(h(b)) → c15(H(ok(b)))
PROPER(g(x0, h(z0))) → c14(G(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(g(x0, g(z0, z1))) → c14(G(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(g(x0, f(z0, z1))) → c14(G(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(g(h(z0), x1)) → c14(G(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(g(g(z0, z1), x1)) → c14(G(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(g(f(z0, z1), x1)) → c14(G(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(g(x0, a)) → c14(G(proper(x0), ok(a)), PROPER(x0))
PROPER(g(x0, b)) → c14(G(proper(x0), ok(b)), PROPER(x0))
PROPER(g(a, x1)) → c14(G(ok(a), proper(x1)), PROPER(x1))
PROPER(g(b, x1)) → c14(G(ok(b), proper(x1)), PROPER(x1))
PROPER(f(x0, h(z0))) → c16(F(proper(x0), h(proper(z0))), PROPER(x0), PROPER(h(z0)))
PROPER(f(x0, g(z0, z1))) → c16(F(proper(x0), g(proper(z0), proper(z1))), PROPER(x0), PROPER(g(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c16(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(h(z0), x1)) → c16(F(h(proper(z0)), proper(x1)), PROPER(h(z0)), PROPER(x1))
PROPER(f(g(z0, z1), x1)) → c16(F(g(proper(z0), proper(z1)), proper(x1)), PROPER(g(z0, z1)), PROPER(x1))
PROPER(f(f(z0, z1), x1)) → c16(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c16(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c16(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c16(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c16(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(h(z0))) → c18(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(g(z0, z1))) → c18(TOP(g(proper(z0), proper(z1))), PROPER(g(z0, z1)))
TOP(mark(f(z0, z1))) → c18(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(ok(h(z0))) → c19(TOP(mark(g(z0, z0))), ACTIVE(h(z0)))
TOP(ok(g(a, z0))) → c19(TOP(mark(f(b, z0))), ACTIVE(g(a, z0)))
TOP(ok(f(z0, z0))) → c19(TOP(mark(h(a))), ACTIVE(f(z0, z0)))
TOP(ok(h(z0))) → c19(TOP(h(active(z0))), ACTIVE(h(z0)))
TOP(ok(g(z0, z1))) → c19(TOP(g(active(z0), z1)), ACTIVE(g(z0, z1)))
TOP(ok(f(z0, z1))) → c19(TOP(f(active(z0), z1)), ACTIVE(f(z0, z1)))

(56) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0))
active(g(a, z0)) → mark(f(b, z0))
active(f(z0, z0)) → mark(h(a))
active(a) → mark(b)
active(h(z0)) → h(active(z0))
active(g(z0, z1)) → g(active(z0), z1)
active(f(z0, z1)) → f(active(z0), z1)
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
g(mark(z0), z1) → mark(g(z0, z1))
g(ok(z0), ok(z1)) → ok(g(z0, z1))
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(h(z0)) → h(proper(z0))
proper(g(z0, z1)) → g(proper(z0), proper(z1))
proper(a) → ok(a)
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
TOP(mark(a)) → c18(TOP(ok(a)))
TOP(mark(b)) → c18(TOP(ok(b)))
TOP(ok(a)) → c19(TOP(mark(b)), ACTIVE(a))

S tuples:

H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
TOP(ok(a)) → c19(TOP(mark(b)), ACTIVE(a))

K tuples:

TOP(mark(a)) → c18(TOP(ok(a)))
TOP(mark(b)) → c18(TOP(ok(b)))

Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

H, G, F, TOP

Compound Symbols:

c7, c8, c9, c10, c11, c12, c18, c19

(57) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 3 of 9 dangling nodes:

TOP(ok(a)) → c19(TOP(mark(b)), ACTIVE(a))
TOP(mark(b)) → c18(TOP(ok(b)))
TOP(mark(a)) → c18(TOP(ok(a)))

(58) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0))
active(g(a, z0)) → mark(f(b, z0))
active(f(z0, z0)) → mark(h(a))
active(a) → mark(b)
active(h(z0)) → h(active(z0))
active(g(z0, z1)) → g(active(z0), z1)
active(f(z0, z1)) → f(active(z0), z1)
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
g(mark(z0), z1) → mark(g(z0, z1))
g(ok(z0), ok(z1)) → ok(g(z0, z1))
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(h(z0)) → h(proper(z0))
proper(g(z0, z1)) → g(proper(z0), proper(z1))
proper(a) → ok(a)
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))

S tuples:

H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))

K tuples:none
Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

H, G, F

Compound Symbols:

c7, c8, c9, c10, c11, c12

(59) 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.

G(ok(z0), ok(z1)) → c10(G(z0, z1))

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

H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))

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

POL(F(x₁, x₂)) = 0
POL(G(x₁, x₂)) = x₁·x₂ + x₁² + x₁²·x₂
POL(H(x₁)) = 0
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(ok(x₁)) = [1] + x₁

(60) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0))
active(g(a, z0)) → mark(f(b, z0))
active(f(z0, z0)) → mark(h(a))
active(a) → mark(b)
active(h(z0)) → h(active(z0))
active(g(z0, z1)) → g(active(z0), z1)
active(f(z0, z1)) → f(active(z0), z1)
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
g(mark(z0), z1) → mark(g(z0, z1))
g(ok(z0), ok(z1)) → ok(g(z0, z1))
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(h(z0)) → h(proper(z0))
proper(g(z0, z1)) → g(proper(z0), proper(z1))
proper(a) → ok(a)
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))

S tuples:

H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))

K tuples:

G(ok(z0), ok(z1)) → c10(G(z0, z1))

Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

H, G, F

Compound Symbols:

c7, c8, c9, c10, c11, c12

(61) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

F(ok(z0), ok(z1)) → c12(F(z0, z1))

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

H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))

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

POL(F(x₁, x₂)) = [3]x₂ + [3]x₂²
POL(G(x₁, x₂)) = 0
POL(H(x₁)) = 0
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = 0
POL(ok(x₁)) = [3] + x₁

(62) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0))
active(g(a, z0)) → mark(f(b, z0))
active(f(z0, z0)) → mark(h(a))
active(a) → mark(b)
active(h(z0)) → h(active(z0))
active(g(z0, z1)) → g(active(z0), z1)
active(f(z0, z1)) → f(active(z0), z1)
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
g(mark(z0), z1) → mark(g(z0, z1))
g(ok(z0), ok(z1)) → ok(g(z0, z1))
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(h(z0)) → h(proper(z0))
proper(g(z0, z1)) → g(proper(z0), proper(z1))
proper(a) → ok(a)
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))

S tuples:

H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))

K tuples:

G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))

Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

H, G, F

Compound Symbols:

c7, c8, c9, c10, c11, c12

(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.

H(ok(z0)) → c8(H(z0))

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

H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))

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

POL(F(x₁, x₂)) = [4]x₁ + [2]x₂
POL(G(x₁, x₂)) = [3]x₁
POL(H(x₁)) = [3]x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = [1] + x₁

(64) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0))
active(g(a, z0)) → mark(f(b, z0))
active(f(z0, z0)) → mark(h(a))
active(a) → mark(b)
active(h(z0)) → h(active(z0))
active(g(z0, z1)) → g(active(z0), z1)
active(f(z0, z1)) → f(active(z0), z1)
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
g(mark(z0), z1) → mark(g(z0, z1))
g(ok(z0), ok(z1)) → ok(g(z0, z1))
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(h(z0)) → h(proper(z0))
proper(g(z0, z1)) → g(proper(z0), proper(z1))
proper(a) → ok(a)
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))

S tuples:

H(mark(z0)) → c7(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))

K tuples:

G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
H(ok(z0)) → c8(H(z0))

Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

H, G, F

Compound Symbols:

c7, c8, c9, c10, c11, c12

(65) 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.

H(mark(z0)) → c7(H(z0))
F(mark(z0), z1) → c11(F(z0, z1))

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

H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))

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

POL(F(x₁, x₂)) = [4]x₁ + [3]x₂
POL(G(x₁, x₂)) = [5]x₂
POL(H(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁

(66) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0))
active(g(a, z0)) → mark(f(b, z0))
active(f(z0, z0)) → mark(h(a))
active(a) → mark(b)
active(h(z0)) → h(active(z0))
active(g(z0, z1)) → g(active(z0), z1)
active(f(z0, z1)) → f(active(z0), z1)
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
g(mark(z0), z1) → mark(g(z0, z1))
g(ok(z0), ok(z1)) → ok(g(z0, z1))
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(h(z0)) → h(proper(z0))
proper(g(z0, z1)) → g(proper(z0), proper(z1))
proper(a) → ok(a)
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))

S tuples:

G(mark(z0), z1) → c9(G(z0, z1))

K tuples:

G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
H(ok(z0)) → c8(H(z0))
H(mark(z0)) → c7(H(z0))
F(mark(z0), z1) → c11(F(z0, z1))

Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

H, G, F

Compound Symbols:

c7, c8, c9, c10, c11, c12

(67) 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), z1) → c9(G(z0, z1))

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

H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))

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

POL(F(x₁, x₂)) = [2]x₂
POL(G(x₁, x₂)) = x₁ + [2]x₂
POL(H(x₁)) = 0
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁

(68) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(h(z0)) → mark(g(z0, z0))
active(g(a, z0)) → mark(f(b, z0))
active(f(z0, z0)) → mark(h(a))
active(a) → mark(b)
active(h(z0)) → h(active(z0))
active(g(z0, z1)) → g(active(z0), z1)
active(f(z0, z1)) → f(active(z0), z1)
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
g(mark(z0), z1) → mark(g(z0, z1))
g(ok(z0), ok(z1)) → ok(g(z0, z1))
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(h(z0)) → h(proper(z0))
proper(g(z0, z1)) → g(proper(z0), proper(z1))
proper(a) → ok(a)
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

H(mark(z0)) → c7(H(z0))
H(ok(z0)) → c8(H(z0))
G(mark(z0), z1) → c9(G(z0, z1))
G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(mark(z0), z1) → c11(F(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))

S tuples:none
K tuples:

G(ok(z0), ok(z1)) → c10(G(z0, z1))
F(ok(z0), ok(z1)) → c12(F(z0, z1))
H(ok(z0)) → c8(H(z0))
H(mark(z0)) → c7(H(z0))
F(mark(z0), z1) → c11(F(z0, z1))
G(mark(z0), z1) → c9(G(z0, z1))

Defined Rule Symbols:

active, h, g, f, proper, top

Defined Pair Symbols:

H, G, F

Compound Symbols:

c7, c8, c9, c10, c11, c12

(69) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(0) Obligation:

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

(2) Obligation:

(3) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

(4) Obligation:

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

(6) Obligation:

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

(8) Obligation:

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

(10) Obligation:

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

(12) Obligation:

(13) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

(14) Obligation:

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

(16) Obligation:

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

(18) Obligation:

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

(20) Obligation:

(21) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

(22) Obligation:

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

(24) Obligation:

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

(26) Obligation:

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

(28) Obligation:

(29) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

(30) Obligation:

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

(32) Obligation:

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

(34) Obligation:

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

(36) Obligation:

(37) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

(38) Obligation:

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

(40) Obligation:

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

(42) Obligation:

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

(44) Obligation:

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

(46) Obligation:

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

(48) Obligation:

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

(50) Obligation:

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

(52) Obligation:

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

(54) Obligation:

(55) CdtUnreachableProof (EQUIVALENT transformation)

(56) Obligation:

(57) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

(58) Obligation:

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

(60) Obligation:

(61) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

(62) Obligation:

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

(64) Obligation:

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

(66) Obligation:

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

(68) Obligation:

(69) SIsEmptyProof (EQUIVALENT transformation)

(70) BOUNDS(O(1), O(1))