Runtime Complexity (innermost) proof of /tmp/tmpSq9BiG/Ex1_Zan97

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 CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID))

↳24 CdtProblem

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

↳26 CdtProblem

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

↳28 CdtProblem

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

↳30 CdtProblem

↳31 CdtUnreachableProof (⇔)

↳32 CdtProblem

↳33 CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication)

↳34 CdtProblem

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

↳36 CdtProblem

↳37 SIsEmptyProof (⇔)

↳38 BOUNDS(O(1), O(1))

(0) Obligation:

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

active(g(X)) → mark(h(X))
active(c) → mark(d)
active(h(d)) → mark(g(c))
proper(g(X)) → g(proper(X))
proper(h(X)) → h(proper(X))
proper(c) → ok(c)
proper(d) → ok(d)
g(ok(X)) → ok(g(X))
h(ok(X)) → ok(h(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(g(z0)) → mark(h(z0))
active(c) → mark(d)
active(h(d)) → mark(g(c))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
proper(c) → ok(c)
proper(d) → ok(d)
g(ok(z0)) → ok(g(z0))
h(ok(z0)) → ok(h(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(g(z0)) → c1(H(z0))
ACTIVE(h(d)) → c3(G(c))
PROPER(g(z0)) → c4(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c5(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(mark(z0)) → c10(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(g(z0)) → c1(H(z0))
ACTIVE(h(d)) → c3(G(c))
PROPER(g(z0)) → c4(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c5(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(mark(z0)) → c10(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, proper, g, h, top

Defined Pair Symbols:

ACTIVE, PROPER, G, H, TOP

Compound Symbols:

c1, c3, c4, c5, c8, c9, c10, c11

(3) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 8 dangling nodes:

ACTIVE(h(d)) → c3(G(c))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(g(z0)) → mark(h(z0))
active(c) → mark(d)
active(h(d)) → mark(g(c))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
proper(c) → ok(c)
proper(d) → ok(d)
g(ok(z0)) → ok(g(z0))
h(ok(z0)) → ok(h(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(g(z0)) → c1(H(z0))
PROPER(g(z0)) → c4(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c5(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(mark(z0)) → c10(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(g(z0)) → c1(H(z0))
PROPER(g(z0)) → c4(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c5(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(mark(z0)) → c10(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, proper, g, h, top

Defined Pair Symbols:

ACTIVE, PROPER, G, H, TOP

Compound Symbols:

c1, c4, c5, c8, c9, c10, c11

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

Use narrowing to replace PROPER(g(z0)) → c4(G(proper(z0)), PROPER(z0)) by

PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c4(G(ok(c)), PROPER(c))
PROPER(g(d)) → c4(G(ok(d)), PROPER(d))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(g(z0)) → mark(h(z0))
active(c) → mark(d)
active(h(d)) → mark(g(c))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
proper(c) → ok(c)
proper(d) → ok(d)
g(ok(z0)) → ok(g(z0))
h(ok(z0)) → ok(h(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(g(z0)) → c1(H(z0))
PROPER(h(z0)) → c5(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(mark(z0)) → c10(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c4(G(ok(c)), PROPER(c))
PROPER(g(d)) → c4(G(ok(d)), PROPER(d))

S tuples:

ACTIVE(g(z0)) → c1(H(z0))
PROPER(h(z0)) → c5(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(mark(z0)) → c10(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c4(G(ok(c)), PROPER(c))
PROPER(g(d)) → c4(G(ok(d)), PROPER(d))

K tuples:none
Defined Rule Symbols:

active, proper, g, h, top

Defined Pair Symbols:

ACTIVE, PROPER, G, H, TOP

Compound Symbols:

c1, c5, c8, c9, c10, c11, 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(g(z0)) → mark(h(z0))
active(c) → mark(d)
active(h(d)) → mark(g(c))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
proper(c) → ok(c)
proper(d) → ok(d)
g(ok(z0)) → ok(g(z0))
h(ok(z0)) → ok(h(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(g(z0)) → c1(H(z0))
PROPER(h(z0)) → c5(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(mark(z0)) → c10(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c2(G(ok(c)))
PROPER(g(c)) → c2(PROPER(c))
PROPER(g(d)) → c2(G(ok(d)))
PROPER(g(d)) → c2(PROPER(d))

S tuples:

ACTIVE(g(z0)) → c1(H(z0))
PROPER(h(z0)) → c5(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(mark(z0)) → c10(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c2(G(ok(c)))
PROPER(g(c)) → c2(PROPER(c))
PROPER(g(d)) → c2(G(ok(d)))
PROPER(g(d)) → c2(PROPER(d))

K tuples:none
Defined Rule Symbols:

active, proper, g, h, top

Defined Pair Symbols:

ACTIVE, PROPER, G, H, TOP

Compound Symbols:

c1, c5, c8, c9, c10, c11, c4, c2

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

Removed 2 trailing tuple parts

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(g(z0)) → mark(h(z0))
active(c) → mark(d)
active(h(d)) → mark(g(c))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
proper(c) → ok(c)
proper(d) → ok(d)
g(ok(z0)) → ok(g(z0))
h(ok(z0)) → ok(h(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(g(z0)) → c1(H(z0))
PROPER(h(z0)) → c5(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(mark(z0)) → c10(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c2(G(ok(c)))
PROPER(g(d)) → c2(G(ok(d)))
PROPER(g(c)) → c2
PROPER(g(d)) → c2

S tuples:

ACTIVE(g(z0)) → c1(H(z0))
PROPER(h(z0)) → c5(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(mark(z0)) → c10(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c2(G(ok(c)))
PROPER(g(d)) → c2(G(ok(d)))
PROPER(g(c)) → c2
PROPER(g(d)) → c2

K tuples:none
Defined Rule Symbols:

active, proper, g, h, top

Defined Pair Symbols:

ACTIVE, PROPER, G, H, TOP

Compound Symbols:

c1, c5, c8, c9, c10, c11, c4, c2, c2

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

Use narrowing to replace PROPER(h(z0)) → c5(H(proper(z0)), PROPER(z0)) by

PROPER(h(g(z0))) → c5(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c5(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(c)) → c5(H(ok(c)), PROPER(c))
PROPER(h(d)) → c5(H(ok(d)), PROPER(d))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(g(z0)) → mark(h(z0))
active(c) → mark(d)
active(h(d)) → mark(g(c))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
proper(c) → ok(c)
proper(d) → ok(d)
g(ok(z0)) → ok(g(z0))
h(ok(z0)) → ok(h(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(g(z0)) → c1(H(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(mark(z0)) → c10(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c2(G(ok(c)))
PROPER(g(d)) → c2(G(ok(d)))
PROPER(g(c)) → c2
PROPER(g(d)) → c2
PROPER(h(g(z0))) → c5(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c5(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(c)) → c5(H(ok(c)), PROPER(c))
PROPER(h(d)) → c5(H(ok(d)), PROPER(d))

S tuples:

ACTIVE(g(z0)) → c1(H(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(mark(z0)) → c10(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c2(G(ok(c)))
PROPER(g(d)) → c2(G(ok(d)))
PROPER(g(c)) → c2
PROPER(g(d)) → c2
PROPER(h(g(z0))) → c5(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c5(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(c)) → c5(H(ok(c)), PROPER(c))
PROPER(h(d)) → c5(H(ok(d)), PROPER(d))

K tuples:none
Defined Rule Symbols:

active, proper, g, h, top

Defined Pair Symbols:

ACTIVE, G, H, TOP, PROPER

Compound Symbols:

c1, c8, c9, c10, c11, c4, c2, c2, c5

(13) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 2 of 15 dangling nodes:

PROPER(g(d)) → c2
PROPER(g(c)) → c2

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(g(z0)) → mark(h(z0))
active(c) → mark(d)
active(h(d)) → mark(g(c))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
proper(c) → ok(c)
proper(d) → ok(d)
g(ok(z0)) → ok(g(z0))
h(ok(z0)) → ok(h(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(g(z0)) → c1(H(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(mark(z0)) → c10(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c2(G(ok(c)))
PROPER(g(d)) → c2(G(ok(d)))
PROPER(h(g(z0))) → c5(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c5(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(c)) → c5(H(ok(c)), PROPER(c))
PROPER(h(d)) → c5(H(ok(d)), PROPER(d))

S tuples:

ACTIVE(g(z0)) → c1(H(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(mark(z0)) → c10(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c2(G(ok(c)))
PROPER(g(d)) → c2(G(ok(d)))
PROPER(h(g(z0))) → c5(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c5(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(c)) → c5(H(ok(c)), PROPER(c))
PROPER(h(d)) → c5(H(ok(d)), PROPER(d))

K tuples:none
Defined Rule Symbols:

active, proper, g, h, top

Defined Pair Symbols:

ACTIVE, G, H, TOP, PROPER

Compound Symbols:

c1, c8, c9, c10, c11, c4, c2, 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(g(z0)) → mark(h(z0))
active(c) → mark(d)
active(h(d)) → mark(g(c))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
proper(c) → ok(c)
proper(d) → ok(d)
g(ok(z0)) → ok(g(z0))
h(ok(z0)) → ok(h(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(g(z0)) → c1(H(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(mark(z0)) → c10(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c2(G(ok(c)))
PROPER(g(d)) → c2(G(ok(d)))
PROPER(h(g(z0))) → c5(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c5(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(c)) → c3(H(ok(c)))
PROPER(h(c)) → c3(PROPER(c))
PROPER(h(d)) → c3(H(ok(d)))
PROPER(h(d)) → c3(PROPER(d))

S tuples:

ACTIVE(g(z0)) → c1(H(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(mark(z0)) → c10(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c2(G(ok(c)))
PROPER(g(d)) → c2(G(ok(d)))
PROPER(h(g(z0))) → c5(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c5(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(c)) → c3(H(ok(c)))
PROPER(h(c)) → c3(PROPER(c))
PROPER(h(d)) → c3(H(ok(d)))
PROPER(h(d)) → c3(PROPER(d))

K tuples:none
Defined Rule Symbols:

active, proper, g, h, top

Defined Pair Symbols:

ACTIVE, G, H, TOP, PROPER

Compound Symbols:

c1, c8, c9, c10, c11, c4, c2, c5, c3

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

Removed 2 trailing tuple parts

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(g(z0)) → mark(h(z0))
active(c) → mark(d)
active(h(d)) → mark(g(c))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
proper(c) → ok(c)
proper(d) → ok(d)
g(ok(z0)) → ok(g(z0))
h(ok(z0)) → ok(h(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(g(z0)) → c1(H(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(mark(z0)) → c10(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c2(G(ok(c)))
PROPER(g(d)) → c2(G(ok(d)))
PROPER(h(g(z0))) → c5(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c5(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(c)) → c3(H(ok(c)))
PROPER(h(d)) → c3(H(ok(d)))
PROPER(h(c)) → c3
PROPER(h(d)) → c3

S tuples:

ACTIVE(g(z0)) → c1(H(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(mark(z0)) → c10(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c2(G(ok(c)))
PROPER(g(d)) → c2(G(ok(d)))
PROPER(h(g(z0))) → c5(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c5(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(c)) → c3(H(ok(c)))
PROPER(h(d)) → c3(H(ok(d)))
PROPER(h(c)) → c3
PROPER(h(d)) → c3

K tuples:none
Defined Rule Symbols:

active, proper, g, h, top

Defined Pair Symbols:

ACTIVE, G, H, TOP, PROPER

Compound Symbols:

c1, c8, c9, c10, c11, c4, c2, c5, c3, c3

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

Use narrowing to replace TOP(mark(z0)) → c10(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(g(z0))) → c10(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(h(z0))) → c10(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(c)) → c10(TOP(ok(c)), PROPER(c))
TOP(mark(d)) → c10(TOP(ok(d)), PROPER(d))

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(g(z0)) → mark(h(z0))
active(c) → mark(d)
active(h(d)) → mark(g(c))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
proper(c) → ok(c)
proper(d) → ok(d)
g(ok(z0)) → ok(g(z0))
h(ok(z0)) → ok(h(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(g(z0)) → c1(H(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c2(G(ok(c)))
PROPER(g(d)) → c2(G(ok(d)))
PROPER(h(g(z0))) → c5(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c5(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(c)) → c3(H(ok(c)))
PROPER(h(d)) → c3(H(ok(d)))
PROPER(h(c)) → c3
PROPER(h(d)) → c3
TOP(mark(g(z0))) → c10(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(h(z0))) → c10(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(c)) → c10(TOP(ok(c)), PROPER(c))
TOP(mark(d)) → c10(TOP(ok(d)), PROPER(d))

S tuples:

ACTIVE(g(z0)) → c1(H(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c2(G(ok(c)))
PROPER(g(d)) → c2(G(ok(d)))
PROPER(h(g(z0))) → c5(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c5(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(c)) → c3(H(ok(c)))
PROPER(h(d)) → c3(H(ok(d)))
PROPER(h(c)) → c3
PROPER(h(d)) → c3
TOP(mark(g(z0))) → c10(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(h(z0))) → c10(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(c)) → c10(TOP(ok(c)), PROPER(c))
TOP(mark(d)) → c10(TOP(ok(d)), PROPER(d))

K tuples:none
Defined Rule Symbols:

active, proper, g, h, top

Defined Pair Symbols:

ACTIVE, G, H, TOP, PROPER

Compound Symbols:

c1, c8, c9, c11, c4, c2, c5, c3, c3, c10

(21) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 2 of 18 dangling nodes:

PROPER(h(d)) → c3
PROPER(h(c)) → c3

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(g(z0)) → mark(h(z0))
active(c) → mark(d)
active(h(d)) → mark(g(c))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
proper(c) → ok(c)
proper(d) → ok(d)
g(ok(z0)) → ok(g(z0))
h(ok(z0)) → ok(h(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(g(z0)) → c1(H(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c2(G(ok(c)))
PROPER(g(d)) → c2(G(ok(d)))
PROPER(h(g(z0))) → c5(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c5(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(c)) → c3(H(ok(c)))
PROPER(h(d)) → c3(H(ok(d)))
TOP(mark(g(z0))) → c10(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(h(z0))) → c10(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(c)) → c10(TOP(ok(c)), PROPER(c))
TOP(mark(d)) → c10(TOP(ok(d)), PROPER(d))

S tuples:

ACTIVE(g(z0)) → c1(H(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c2(G(ok(c)))
PROPER(g(d)) → c2(G(ok(d)))
PROPER(h(g(z0))) → c5(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c5(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(c)) → c3(H(ok(c)))
PROPER(h(d)) → c3(H(ok(d)))
TOP(mark(g(z0))) → c10(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(h(z0))) → c10(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(c)) → c10(TOP(ok(c)), PROPER(c))
TOP(mark(d)) → c10(TOP(ok(d)), PROPER(d))

K tuples:none
Defined Rule Symbols:

active, proper, g, h, top

Defined Pair Symbols:

ACTIVE, G, H, TOP, PROPER

Compound Symbols:

c1, c8, c9, c11, c4, c2, c5, c3, c10

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

Removed 2 trailing tuple parts

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(g(z0)) → mark(h(z0))
active(c) → mark(d)
active(h(d)) → mark(g(c))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
proper(c) → ok(c)
proper(d) → ok(d)
g(ok(z0)) → ok(g(z0))
h(ok(z0)) → ok(h(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(g(z0)) → c1(H(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c2(G(ok(c)))
PROPER(g(d)) → c2(G(ok(d)))
PROPER(h(g(z0))) → c5(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c5(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(c)) → c3(H(ok(c)))
PROPER(h(d)) → c3(H(ok(d)))
TOP(mark(g(z0))) → c10(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(h(z0))) → c10(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(c)) → c10(TOP(ok(c)))
TOP(mark(d)) → c10(TOP(ok(d)))

S tuples:

ACTIVE(g(z0)) → c1(H(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c2(G(ok(c)))
PROPER(g(d)) → c2(G(ok(d)))
PROPER(h(g(z0))) → c5(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c5(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(c)) → c3(H(ok(c)))
PROPER(h(d)) → c3(H(ok(d)))
TOP(mark(g(z0))) → c10(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(h(z0))) → c10(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(c)) → c10(TOP(ok(c)))
TOP(mark(d)) → c10(TOP(ok(d)))

K tuples:none
Defined Rule Symbols:

active, proper, g, h, top

Defined Pair Symbols:

ACTIVE, G, H, TOP, PROPER

Compound Symbols:

c1, c8, c9, c11, c4, c2, c5, c3, c10, c10

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

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

TOP(mark(c)) → c10(TOP(ok(c)))

We considered the (Usable) Rules:

proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
proper(c) → ok(c)
proper(d) → ok(d)
h(ok(z0)) → ok(h(z0))
g(ok(z0)) → ok(g(z0))
active(g(z0)) → mark(h(z0))
active(c) → mark(d)
active(h(d)) → mark(g(c))

And the Tuples:

ACTIVE(g(z0)) → c1(H(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c2(G(ok(c)))
PROPER(g(d)) → c2(G(ok(d)))
PROPER(h(g(z0))) → c5(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c5(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(c)) → c3(H(ok(c)))
PROPER(h(d)) → c3(H(ok(d)))
TOP(mark(g(z0))) → c10(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(h(z0))) → c10(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(c)) → c10(TOP(ok(c)))
TOP(mark(d)) → c10(TOP(ok(d)))

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

POL(ACTIVE(x₁)) = 0
POL(G(x₁)) = 0
POL(H(x₁)) = 0
POL(PROPER(x₁)) = 0
POL(TOP(x₁)) = [4]x₁
POL(active(x₁)) = 0
POL(c) = [4]
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c10(x₁, x₂)) = x₁ + x₂
POL(c11(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(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(d) = 0
POL(g(x₁)) = 0
POL(h(x₁)) = 0
POL(mark(x₁)) = x₁
POL(ok(x₁)) = 0
POL(proper(x₁)) = 0

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(g(z0)) → mark(h(z0))
active(c) → mark(d)
active(h(d)) → mark(g(c))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
proper(c) → ok(c)
proper(d) → ok(d)
g(ok(z0)) → ok(g(z0))
h(ok(z0)) → ok(h(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(g(z0)) → c1(H(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c2(G(ok(c)))
PROPER(g(d)) → c2(G(ok(d)))
PROPER(h(g(z0))) → c5(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c5(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(c)) → c3(H(ok(c)))
PROPER(h(d)) → c3(H(ok(d)))
TOP(mark(g(z0))) → c10(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(h(z0))) → c10(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(c)) → c10(TOP(ok(c)))
TOP(mark(d)) → c10(TOP(ok(d)))

S tuples:

ACTIVE(g(z0)) → c1(H(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c2(G(ok(c)))
PROPER(g(d)) → c2(G(ok(d)))
PROPER(h(g(z0))) → c5(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c5(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(c)) → c3(H(ok(c)))
PROPER(h(d)) → c3(H(ok(d)))
TOP(mark(g(z0))) → c10(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(h(z0))) → c10(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(d)) → c10(TOP(ok(d)))

K tuples:

TOP(mark(c)) → c10(TOP(ok(c)))

Defined Rule Symbols:

active, proper, g, h, top

Defined Pair Symbols:

ACTIVE, G, H, TOP, PROPER

Compound Symbols:

c1, c8, c9, c11, c4, c2, c5, c3, c10, c10

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

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

TOP(mark(d)) → c10(TOP(ok(d)))

We considered the (Usable) Rules:

proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
proper(c) → ok(c)
proper(d) → ok(d)
h(ok(z0)) → ok(h(z0))
g(ok(z0)) → ok(g(z0))
active(g(z0)) → mark(h(z0))
active(c) → mark(d)
active(h(d)) → mark(g(c))

And the Tuples:

ACTIVE(g(z0)) → c1(H(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c2(G(ok(c)))
PROPER(g(d)) → c2(G(ok(d)))
PROPER(h(g(z0))) → c5(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c5(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(c)) → c3(H(ok(c)))
PROPER(h(d)) → c3(H(ok(d)))
TOP(mark(g(z0))) → c10(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(h(z0))) → c10(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(c)) → c10(TOP(ok(c)))
TOP(mark(d)) → c10(TOP(ok(d)))

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

POL(ACTIVE(x₁)) = 0
POL(G(x₁)) = 0
POL(H(x₁)) = 0
POL(PROPER(x₁)) = 0
POL(TOP(x₁)) = x₁
POL(active(x₁)) = x₁
POL(c) = [2]
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c10(x₁, x₂)) = x₁ + x₂
POL(c11(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(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(d) = 0
POL(g(x₁)) = [2]
POL(h(x₁)) = [2]
POL(mark(x₁)) = [2]
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(g(z0)) → mark(h(z0))
active(c) → mark(d)
active(h(d)) → mark(g(c))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
proper(c) → ok(c)
proper(d) → ok(d)
g(ok(z0)) → ok(g(z0))
h(ok(z0)) → ok(h(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(g(z0)) → c1(H(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c2(G(ok(c)))
PROPER(g(d)) → c2(G(ok(d)))
PROPER(h(g(z0))) → c5(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c5(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(c)) → c3(H(ok(c)))
PROPER(h(d)) → c3(H(ok(d)))
TOP(mark(g(z0))) → c10(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(h(z0))) → c10(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(c)) → c10(TOP(ok(c)))
TOP(mark(d)) → c10(TOP(ok(d)))

S tuples:

ACTIVE(g(z0)) → c1(H(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c2(G(ok(c)))
PROPER(g(d)) → c2(G(ok(d)))
PROPER(h(g(z0))) → c5(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c5(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(c)) → c3(H(ok(c)))
PROPER(h(d)) → c3(H(ok(d)))
TOP(mark(g(z0))) → c10(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(h(z0))) → c10(TOP(h(proper(z0))), PROPER(h(z0)))

K tuples:

TOP(mark(c)) → c10(TOP(ok(c)))
TOP(mark(d)) → c10(TOP(ok(d)))

Defined Rule Symbols:

active, proper, g, h, top

Defined Pair Symbols:

ACTIVE, G, H, TOP, PROPER

Compound Symbols:

c1, c8, c9, c11, c4, c2, c5, c3, c10, c10

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

Use narrowing to replace TOP(ok(z0)) → c11(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(g(z0))) → c11(TOP(mark(h(z0))), ACTIVE(g(z0)))
TOP(ok(c)) → c11(TOP(mark(d)), ACTIVE(c))
TOP(ok(h(d))) → c11(TOP(mark(g(c))), ACTIVE(h(d)))

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(g(z0)) → mark(h(z0))
active(c) → mark(d)
active(h(d)) → mark(g(c))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
proper(c) → ok(c)
proper(d) → ok(d)
g(ok(z0)) → ok(g(z0))
h(ok(z0)) → ok(h(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(g(z0)) → c1(H(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c2(G(ok(c)))
PROPER(g(d)) → c2(G(ok(d)))
PROPER(h(g(z0))) → c5(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c5(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(c)) → c3(H(ok(c)))
PROPER(h(d)) → c3(H(ok(d)))
TOP(mark(g(z0))) → c10(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(h(z0))) → c10(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(c)) → c10(TOP(ok(c)))
TOP(mark(d)) → c10(TOP(ok(d)))
TOP(ok(g(z0))) → c11(TOP(mark(h(z0))), ACTIVE(g(z0)))
TOP(ok(c)) → c11(TOP(mark(d)), ACTIVE(c))
TOP(ok(h(d))) → c11(TOP(mark(g(c))), ACTIVE(h(d)))

S tuples:

ACTIVE(g(z0)) → c1(H(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c2(G(ok(c)))
PROPER(g(d)) → c2(G(ok(d)))
PROPER(h(g(z0))) → c5(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c5(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(c)) → c3(H(ok(c)))
PROPER(h(d)) → c3(H(ok(d)))
TOP(mark(g(z0))) → c10(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(h(z0))) → c10(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(ok(g(z0))) → c11(TOP(mark(h(z0))), ACTIVE(g(z0)))
TOP(ok(c)) → c11(TOP(mark(d)), ACTIVE(c))
TOP(ok(h(d))) → c11(TOP(mark(g(c))), ACTIVE(h(d)))

K tuples:

TOP(mark(c)) → c10(TOP(ok(c)))
TOP(mark(d)) → c10(TOP(ok(d)))

Defined Rule Symbols:

active, proper, g, h, top

Defined Pair Symbols:

ACTIVE, G, H, PROPER, TOP

Compound Symbols:

c1, c8, c9, c4, c2, c5, c3, c10, c10, c11

(31) CdtUnreachableProof (EQUIVALENT transformation)

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

ACTIVE(g(z0)) → c1(H(z0))
PROPER(g(g(z0))) → c4(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c4(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c2(G(ok(c)))
PROPER(g(d)) → c2(G(ok(d)))
PROPER(h(g(z0))) → c5(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c5(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(c)) → c3(H(ok(c)))
PROPER(h(d)) → c3(H(ok(d)))
TOP(mark(g(z0))) → c10(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(h(z0))) → c10(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(ok(g(z0))) → c11(TOP(mark(h(z0))), ACTIVE(g(z0)))
TOP(ok(h(d))) → c11(TOP(mark(g(c))), ACTIVE(h(d)))

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(g(z0)) → mark(h(z0))
active(c) → mark(d)
active(h(d)) → mark(g(c))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
proper(c) → ok(c)
proper(d) → ok(d)
g(ok(z0)) → ok(g(z0))
h(ok(z0)) → ok(h(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(mark(c)) → c10(TOP(ok(c)))
TOP(mark(d)) → c10(TOP(ok(d)))
TOP(ok(c)) → c11(TOP(mark(d)), ACTIVE(c))

S tuples:

G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
TOP(ok(c)) → c11(TOP(mark(d)), ACTIVE(c))

K tuples:

TOP(mark(c)) → c10(TOP(ok(c)))
TOP(mark(d)) → c10(TOP(ok(d)))

Defined Rule Symbols:

active, proper, g, h, top

Defined Pair Symbols:

G, H, TOP

Compound Symbols:

c8, c9, c10, c11

(33) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 3 of 5 dangling nodes:

TOP(mark(d)) → c10(TOP(ok(d)))
TOP(mark(c)) → c10(TOP(ok(c)))
TOP(ok(c)) → c11(TOP(mark(d)), ACTIVE(c))

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(g(z0)) → mark(h(z0))
active(c) → mark(d)
active(h(d)) → mark(g(c))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
proper(c) → ok(c)
proper(d) → ok(d)
g(ok(z0)) → ok(g(z0))
h(ok(z0)) → ok(h(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))

S tuples:

G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))

K tuples:none
Defined Rule Symbols:

active, proper, g, h, top

Defined Pair Symbols:

G, H

Compound Symbols:

c8, c9

(35) 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)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))

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

G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))

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

POL(G(x₁)) = x₁²
POL(H(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(ok(x₁)) = [1] + x₁

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(g(z0)) → mark(h(z0))
active(c) → mark(d)
active(h(d)) → mark(g(c))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
proper(c) → ok(c)
proper(d) → ok(d)
g(ok(z0)) → ok(g(z0))
h(ok(z0)) → ok(h(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))

S tuples:none
K tuples:

G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))

Defined Rule Symbols:

active, proper, g, h, top

Defined Pair Symbols:

G, H

Compound Symbols:

c8, c9

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

(24) Obligation:

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

(26) Obligation:

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

(28) Obligation:

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

(30) Obligation:

(31) CdtUnreachableProof (EQUIVALENT transformation)

(32) Obligation:

(33) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

(34) Obligation:

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

(36) Obligation:

(37) SIsEmptyProof (EQUIVALENT transformation)

(38) BOUNDS(O(1), O(1))