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

0 CpxTRS
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)
↳2 CdtProblem
    ↳3 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳4 CdtProblem
    ↳5 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 11 ms)
    ↳6 CdtProblem
    ↳7 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳8 CdtProblem
    ↳9 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 16 ms)
    ↳10 CdtProblem
    ↳11 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳12 CdtProblem
    ↳13 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳14 CdtProblem
    ↳15 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳16 CdtProblem
    ↳17 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 192 ms)
    ↳18 CdtProblem
    ↳19 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 221 ms)
    ↳20 CdtProblem
    ↳21 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳22 CdtProblem
    ↳23 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳24 CdtProblem
    ↳25 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 201 ms)
    ↳26 CdtProblem
    ↳27 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 508 ms)
    ↳28 CdtProblem
    ↳29 CdtKnowledgeProof (⇔, 0 ms)
    ↳30 CdtProblem
    ↳31 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 193 ms)
    ↳32 CdtProblem
    ↳33 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳34 CdtProblem
    ↳35 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳36 CdtProblem
    ↳37 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳38 CdtProblem
    ↳39 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳40 CdtProblem
    ↳41 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳42 CdtProblem
    ↳43 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 543 ms)
    ↳44 CdtProblem
↳45 CpxTrsMatchBoundsProof (⇔, 0 ms)
↳46 BOUNDS(O(1), O(n^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) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing 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))
PROPER(g(x0)) → c4

(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))
PROPER(g(x0)) → c4
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))
PROPER(g(x0)) → c4
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, c4

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

Removed 1 trailing nodes:

PROPER(g(x0)) → c4

(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)) → 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

(9) 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))
PROPER(h(x0)) → c5

(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))
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))
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))
PROPER(h(x0)) → c5
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)) → c4(G(ok(c)), PROPER(c))
PROPER(g(d)) → c4(G(ok(d)), PROPER(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))
PROPER(h(x0)) → c5
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, c5, c5

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

Removed 1 trailing nodes:

PROPER(h(x0)) → c5

(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)) → c4(G(ok(c)), PROPER(c))
PROPER(g(d)) → c4(G(ok(d)), PROPER(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)) → c4(G(ok(c)), PROPER(c))
PROPER(g(d)) → c4(G(ok(d)), PROPER(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, c5

(13) 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))
TOP(mark(x0)) → c10

(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(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))
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))
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))
TOP(mark(x0)) → c10
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)) → c4(G(ok(c)), PROPER(c))
PROPER(g(d)) → c4(G(ok(d)), PROPER(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))
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))
TOP(mark(x0)) → c10
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, c5, c10, c10

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

Removed 1 trailing nodes:

TOP(mark(x0)) → c10

(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(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))
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))
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)) → c4(G(ok(c)), PROPER(c))
PROPER(g(d)) → c4(G(ok(d)), PROPER(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))
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, c5, c10

(17) 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)), PROPER(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)) → c4(G(ok(c)), PROPER(c))
PROPER(g(d)) → c4(G(ok(d)), PROPER(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))
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))
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)) = 0   
POL(c) = [2]   
POL(c1(x1)) = x1   
POL(c10(x1, x2)) = x1 + x2   
POL(c11(x1, x2)) = x1 + x2   
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   

(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(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))
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))
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)) → c4(G(ok(c)), PROPER(c))
PROPER(g(d)) → c4(G(ok(d)), PROPER(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))
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)), PROPER(d))
K tuples:

TOP(mark(c)) → c10(TOP(ok(c)), PROPER(c))
Defined Rule Symbols:

active, proper, g, h, top

Defined Pair Symbols:

ACTIVE, G, H, TOP, PROPER

Compound Symbols:

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

(19) 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)), PROPER(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)) → c4(G(ok(c)), PROPER(c))
PROPER(g(d)) → c4(G(ok(d)), PROPER(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))
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))
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) = [1]   
POL(c1(x1)) = x1   
POL(c10(x1, x2)) = x1 + x2   
POL(c11(x1, x2)) = x1 + x2   
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)) = [1]   
POL(h(x1)) = [1]   
POL(mark(x1)) = [1]   
POL(ok(x1)) = x1   
POL(proper(x1)) = 0   

(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)) → c4(G(ok(c)), PROPER(c))
PROPER(g(d)) → c4(G(ok(d)), PROPER(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))
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)) → c4(G(ok(c)), PROPER(c))
PROPER(g(d)) → c4(G(ok(d)), PROPER(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))
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)), PROPER(c))
TOP(mark(d)) → c10(TOP(ok(d)), PROPER(d))
Defined Rule Symbols:

active, proper, g, h, top

Defined Pair Symbols:

ACTIVE, G, H, TOP, PROPER

Compound Symbols:

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

(21) 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)))
TOP(ok(x0)) → c11

(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))
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))
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))
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))
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)))
TOP(ok(x0)) → c11
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)) → c4(G(ok(c)), PROPER(c))
PROPER(g(d)) → c4(G(ok(d)), PROPER(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))
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)))
TOP(ok(x0)) → c11
K tuples:

TOP(mark(c)) → c10(TOP(ok(c)), PROPER(c))
TOP(mark(d)) → c10(TOP(ok(d)), PROPER(d))
Defined Rule Symbols:

active, proper, g, h, top

Defined Pair Symbols:

ACTIVE, G, H, PROPER, TOP

Compound Symbols:

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

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

Removed 4 trailing nodes:

TOP(ok(c)) → c11(TOP(mark(d)), ACTIVE(c))
TOP(mark(d)) → c10(TOP(ok(d)), PROPER(d))
TOP(ok(x0)) → c11
TOP(mark(c)) → c10(TOP(ok(c)), PROPER(c))

(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))
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))
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))
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)))
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)) → c4(G(ok(c)), PROPER(c))
PROPER(g(d)) → c4(G(ok(d)), PROPER(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))
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)))
K tuples:none
Defined Rule Symbols:

active, proper, g, h, top

Defined Pair Symbols:

ACTIVE, G, H, PROPER, TOP

Compound Symbols:

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

(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(ok(h(d))) → c11(TOP(mark(g(c))), ACTIVE(h(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))
And the 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)) → c4(G(ok(c)), PROPER(c))
PROPER(g(d)) → c4(G(ok(d)), PROPER(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))
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)))
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)) = [2]x1   
POL(c) = 0   
POL(c1(x1)) = x1   
POL(c10(x1, x2)) = x1 + x2   
POL(c11(x1, x2)) = x1 + x2   
POL(c4(x1, x2)) = x1 + x2   
POL(c5(x1, x2)) = x1 + x2   
POL(c8(x1)) = x1   
POL(c9(x1)) = x1   
POL(d) = [2]   
POL(g(x1)) = [4]x1   
POL(h(x1)) = [4]x1   
POL(mark(x1)) = x1   
POL(ok(x1)) = x1   
POL(proper(x1)) = x1   

(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))
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))
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))
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)))
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)) → c4(G(ok(c)), PROPER(c))
PROPER(g(d)) → c4(G(ok(d)), PROPER(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))
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)))
K tuples:

TOP(ok(h(d))) → c11(TOP(mark(g(c))), ACTIVE(h(d)))
Defined Rule Symbols:

active, proper, g, h, top

Defined Pair Symbols:

ACTIVE, G, H, PROPER, TOP

Compound Symbols:

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

(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(ok(g(z0))) → c11(TOP(mark(h(z0))), ACTIVE(g(z0)))
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))
And the 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)) → c4(G(ok(c)), PROPER(c))
PROPER(g(d)) → c4(G(ok(d)), PROPER(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))
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)))
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(c) = 0   
POL(c1(x1)) = x1   
POL(c10(x1, x2)) = x1 + x2   
POL(c11(x1, x2)) = x1 + x2   
POL(c4(x1, x2)) = x1 + x2   
POL(c5(x1, x2)) = x1 + x2   
POL(c8(x1)) = x1   
POL(c9(x1)) = x1   
POL(d) = [1]   
POL(g(x1)) = [1] + [4]x1   
POL(h(x1)) = [4]x1   
POL(mark(x1)) = x1   
POL(ok(x1)) = x1   
POL(proper(x1)) = x1   

(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))
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))
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))
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)))
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)) → c4(G(ok(c)), PROPER(c))
PROPER(g(d)) → c4(G(ok(d)), PROPER(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))
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(ok(h(d))) → c11(TOP(mark(g(c))), ACTIVE(h(d)))
TOP(ok(g(z0))) → c11(TOP(mark(h(z0))), ACTIVE(g(z0)))
Defined Rule Symbols:

active, proper, g, h, top

Defined Pair Symbols:

ACTIVE, G, H, PROPER, TOP

Compound Symbols:

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

(29) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

ACTIVE(g(z0)) → c1(H(z0))

(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)) → c4(G(ok(c)), PROPER(c))
PROPER(g(d)) → c4(G(ok(d)), PROPER(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))
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)))
S tuples:

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)) → c4(G(ok(c)), PROPER(c))
PROPER(g(d)) → c4(G(ok(d)), PROPER(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))
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(ok(h(d))) → c11(TOP(mark(g(c))), ACTIVE(h(d)))
TOP(ok(g(z0))) → c11(TOP(mark(h(z0))), ACTIVE(g(z0)))
ACTIVE(g(z0)) → c1(H(z0))
Defined Rule Symbols:

active, proper, g, h, top

Defined Pair Symbols:

ACTIVE, G, H, PROPER, TOP

Compound Symbols:

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

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

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

TOP(mark(g(z0))) → c10(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(h(z0))) → c10(TOP(h(proper(z0))), PROPER(h(z0)))
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))
And the 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)) → c4(G(ok(c)), PROPER(c))
PROPER(g(d)) → c4(G(ok(d)), PROPER(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))
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)))
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(c) = 0   
POL(c1(x1)) = x1   
POL(c10(x1, x2)) = x1 + x2   
POL(c11(x1, x2)) = x1 + x2   
POL(c4(x1, x2)) = x1 + x2   
POL(c5(x1, x2)) = x1 + x2   
POL(c8(x1)) = x1   
POL(c9(x1)) = x1   
POL(d) = [2]   
POL(g(x1)) = [1] + [2]x1   
POL(h(x1)) = [2]x1   
POL(mark(x1)) = [1] + x1   
POL(ok(x1)) = x1   
POL(proper(x1)) = x1   

(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:

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)) → c4(G(ok(c)), PROPER(c))
PROPER(g(d)) → c4(G(ok(d)), PROPER(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))
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)))
S tuples:

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)) → c4(G(ok(c)), PROPER(c))
PROPER(g(d)) → c4(G(ok(d)), PROPER(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:

TOP(ok(h(d))) → c11(TOP(mark(g(c))), ACTIVE(h(d)))
TOP(ok(g(z0))) → c11(TOP(mark(h(z0))), ACTIVE(g(z0)))
ACTIVE(g(z0)) → c1(H(z0))
TOP(mark(g(z0))) → c10(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(h(z0))) → c10(TOP(h(proper(z0))), PROPER(h(z0)))
Defined Rule Symbols:

active, proper, g, h, top

Defined Pair Symbols:

ACTIVE, G, H, PROPER, TOP

Compound Symbols:

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

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

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

PROPER(g(g(g(z0)))) → c4(G(g(g(proper(z0)))), PROPER(g(g(z0))))
PROPER(g(g(h(z0)))) → c4(G(g(h(proper(z0)))), PROPER(g(h(z0))))
PROPER(g(g(c))) → c4(G(g(ok(c))), PROPER(g(c)))
PROPER(g(g(d))) → c4(G(g(ok(d))), PROPER(g(d)))
PROPER(g(g(x0))) → c4

(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:

ACTIVE(g(z0)) → c1(H(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(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))
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))
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)))
PROPER(g(g(g(z0)))) → c4(G(g(g(proper(z0)))), PROPER(g(g(z0))))
PROPER(g(g(h(z0)))) → c4(G(g(h(proper(z0)))), PROPER(g(h(z0))))
PROPER(g(g(c))) → c4(G(g(ok(c))), PROPER(g(c)))
PROPER(g(g(d))) → c4(G(g(ok(d))), PROPER(g(d)))
PROPER(g(g(x0))) → c4
S tuples:

G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(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))
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))
PROPER(g(g(g(z0)))) → c4(G(g(g(proper(z0)))), PROPER(g(g(z0))))
PROPER(g(g(h(z0)))) → c4(G(g(h(proper(z0)))), PROPER(g(h(z0))))
PROPER(g(g(c))) → c4(G(g(ok(c))), PROPER(g(c)))
PROPER(g(g(d))) → c4(G(g(ok(d))), PROPER(g(d)))
PROPER(g(g(x0))) → c4
K tuples:

TOP(ok(h(d))) → c11(TOP(mark(g(c))), ACTIVE(h(d)))
TOP(ok(g(z0))) → c11(TOP(mark(h(z0))), ACTIVE(g(z0)))
ACTIVE(g(z0)) → c1(H(z0))
TOP(mark(g(z0))) → c10(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(h(z0))) → c10(TOP(h(proper(z0))), PROPER(h(z0)))
Defined Rule Symbols:

active, proper, g, h, top

Defined Pair Symbols:

ACTIVE, G, H, PROPER, TOP

Compound Symbols:

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

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

Removed 1 trailing nodes:

PROPER(g(g(x0))) → c4

(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:

ACTIVE(g(z0)) → c1(H(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(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))
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))
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)))
PROPER(g(g(g(z0)))) → c4(G(g(g(proper(z0)))), PROPER(g(g(z0))))
PROPER(g(g(h(z0)))) → c4(G(g(h(proper(z0)))), PROPER(g(h(z0))))
PROPER(g(g(c))) → c4(G(g(ok(c))), PROPER(g(c)))
PROPER(g(g(d))) → c4(G(g(ok(d))), PROPER(g(d)))
S tuples:

G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(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))
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))
PROPER(g(g(g(z0)))) → c4(G(g(g(proper(z0)))), PROPER(g(g(z0))))
PROPER(g(g(h(z0)))) → c4(G(g(h(proper(z0)))), PROPER(g(h(z0))))
PROPER(g(g(c))) → c4(G(g(ok(c))), PROPER(g(c)))
PROPER(g(g(d))) → c4(G(g(ok(d))), PROPER(g(d)))
K tuples:

TOP(ok(h(d))) → c11(TOP(mark(g(c))), ACTIVE(h(d)))
TOP(ok(g(z0))) → c11(TOP(mark(h(z0))), ACTIVE(g(z0)))
ACTIVE(g(z0)) → c1(H(z0))
TOP(mark(g(z0))) → c10(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(h(z0))) → c10(TOP(h(proper(z0))), PROPER(h(z0)))
Defined Rule Symbols:

active, proper, g, h, top

Defined Pair Symbols:

ACTIVE, G, H, PROPER, TOP

Compound Symbols:

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

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

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

PROPER(g(h(g(z0)))) → c4(G(h(g(proper(z0)))), PROPER(h(g(z0))))
PROPER(g(h(h(z0)))) → c4(G(h(h(proper(z0)))), PROPER(h(h(z0))))
PROPER(g(h(c))) → c4(G(h(ok(c))), PROPER(h(c)))
PROPER(g(h(d))) → c4(G(h(ok(d))), PROPER(h(d)))
PROPER(g(h(x0))) → c4

(38) 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(c)) → c4(G(ok(c)), PROPER(c))
PROPER(g(d)) → c4(G(ok(d)), PROPER(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))
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)))
PROPER(g(g(g(z0)))) → c4(G(g(g(proper(z0)))), PROPER(g(g(z0))))
PROPER(g(g(h(z0)))) → c4(G(g(h(proper(z0)))), PROPER(g(h(z0))))
PROPER(g(g(c))) → c4(G(g(ok(c))), PROPER(g(c)))
PROPER(g(g(d))) → c4(G(g(ok(d))), PROPER(g(d)))
PROPER(g(h(g(z0)))) → c4(G(h(g(proper(z0)))), PROPER(h(g(z0))))
PROPER(g(h(h(z0)))) → c4(G(h(h(proper(z0)))), PROPER(h(h(z0))))
PROPER(g(h(c))) → c4(G(h(ok(c))), PROPER(h(c)))
PROPER(g(h(d))) → c4(G(h(ok(d))), PROPER(h(d)))
PROPER(g(h(x0))) → c4
S tuples:

G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
PROPER(g(c)) → c4(G(ok(c)), PROPER(c))
PROPER(g(d)) → c4(G(ok(d)), PROPER(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))
PROPER(g(g(g(z0)))) → c4(G(g(g(proper(z0)))), PROPER(g(g(z0))))
PROPER(g(g(h(z0)))) → c4(G(g(h(proper(z0)))), PROPER(g(h(z0))))
PROPER(g(g(c))) → c4(G(g(ok(c))), PROPER(g(c)))
PROPER(g(g(d))) → c4(G(g(ok(d))), PROPER(g(d)))
PROPER(g(h(g(z0)))) → c4(G(h(g(proper(z0)))), PROPER(h(g(z0))))
PROPER(g(h(h(z0)))) → c4(G(h(h(proper(z0)))), PROPER(h(h(z0))))
PROPER(g(h(c))) → c4(G(h(ok(c))), PROPER(h(c)))
PROPER(g(h(d))) → c4(G(h(ok(d))), PROPER(h(d)))
PROPER(g(h(x0))) → c4
K tuples:

TOP(ok(h(d))) → c11(TOP(mark(g(c))), ACTIVE(h(d)))
TOP(ok(g(z0))) → c11(TOP(mark(h(z0))), ACTIVE(g(z0)))
ACTIVE(g(z0)) → c1(H(z0))
TOP(mark(g(z0))) → c10(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(h(z0))) → c10(TOP(h(proper(z0))), PROPER(h(z0)))
Defined Rule Symbols:

active, proper, g, h, top

Defined Pair Symbols:

ACTIVE, G, H, PROPER, TOP

Compound Symbols:

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

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

Removed 1 trailing nodes:

PROPER(g(h(x0))) → c4

(40) 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(c)) → c4(G(ok(c)), PROPER(c))
PROPER(g(d)) → c4(G(ok(d)), PROPER(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))
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)))
PROPER(g(g(g(z0)))) → c4(G(g(g(proper(z0)))), PROPER(g(g(z0))))
PROPER(g(g(h(z0)))) → c4(G(g(h(proper(z0)))), PROPER(g(h(z0))))
PROPER(g(g(c))) → c4(G(g(ok(c))), PROPER(g(c)))
PROPER(g(g(d))) → c4(G(g(ok(d))), PROPER(g(d)))
PROPER(g(h(g(z0)))) → c4(G(h(g(proper(z0)))), PROPER(h(g(z0))))
PROPER(g(h(h(z0)))) → c4(G(h(h(proper(z0)))), PROPER(h(h(z0))))
PROPER(g(h(c))) → c4(G(h(ok(c))), PROPER(h(c)))
PROPER(g(h(d))) → c4(G(h(ok(d))), PROPER(h(d)))
S tuples:

G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
PROPER(g(c)) → c4(G(ok(c)), PROPER(c))
PROPER(g(d)) → c4(G(ok(d)), PROPER(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))
PROPER(g(g(g(z0)))) → c4(G(g(g(proper(z0)))), PROPER(g(g(z0))))
PROPER(g(g(h(z0)))) → c4(G(g(h(proper(z0)))), PROPER(g(h(z0))))
PROPER(g(g(c))) → c4(G(g(ok(c))), PROPER(g(c)))
PROPER(g(g(d))) → c4(G(g(ok(d))), PROPER(g(d)))
PROPER(g(h(g(z0)))) → c4(G(h(g(proper(z0)))), PROPER(h(g(z0))))
PROPER(g(h(h(z0)))) → c4(G(h(h(proper(z0)))), PROPER(h(h(z0))))
PROPER(g(h(c))) → c4(G(h(ok(c))), PROPER(h(c)))
PROPER(g(h(d))) → c4(G(h(ok(d))), PROPER(h(d)))
K tuples:

TOP(ok(h(d))) → c11(TOP(mark(g(c))), ACTIVE(h(d)))
TOP(ok(g(z0))) → c11(TOP(mark(h(z0))), ACTIVE(g(z0)))
ACTIVE(g(z0)) → c1(H(z0))
TOP(mark(g(z0))) → c10(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(h(z0))) → c10(TOP(h(proper(z0))), PROPER(h(z0)))
Defined Rule Symbols:

active, proper, g, h, top

Defined Pair Symbols:

ACTIVE, G, H, PROPER, TOP

Compound Symbols:

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

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

Use narrowing to replace PROPER(g(c)) → c4(G(ok(c)), PROPER(c)) by

PROPER(g(c)) → c4(G(ok(c)))

(42) 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(d)) → c4(G(ok(d)), PROPER(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))
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)))
PROPER(g(g(g(z0)))) → c4(G(g(g(proper(z0)))), PROPER(g(g(z0))))
PROPER(g(g(h(z0)))) → c4(G(g(h(proper(z0)))), PROPER(g(h(z0))))
PROPER(g(g(c))) → c4(G(g(ok(c))), PROPER(g(c)))
PROPER(g(g(d))) → c4(G(g(ok(d))), PROPER(g(d)))
PROPER(g(h(g(z0)))) → c4(G(h(g(proper(z0)))), PROPER(h(g(z0))))
PROPER(g(h(h(z0)))) → c4(G(h(h(proper(z0)))), PROPER(h(h(z0))))
PROPER(g(h(c))) → c4(G(h(ok(c))), PROPER(h(c)))
PROPER(g(h(d))) → c4(G(h(ok(d))), PROPER(h(d)))
PROPER(g(c)) → c4(G(ok(c)))
S tuples:

G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
PROPER(g(d)) → c4(G(ok(d)), PROPER(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))
PROPER(g(g(g(z0)))) → c4(G(g(g(proper(z0)))), PROPER(g(g(z0))))
PROPER(g(g(h(z0)))) → c4(G(g(h(proper(z0)))), PROPER(g(h(z0))))
PROPER(g(g(c))) → c4(G(g(ok(c))), PROPER(g(c)))
PROPER(g(g(d))) → c4(G(g(ok(d))), PROPER(g(d)))
PROPER(g(h(g(z0)))) → c4(G(h(g(proper(z0)))), PROPER(h(g(z0))))
PROPER(g(h(h(z0)))) → c4(G(h(h(proper(z0)))), PROPER(h(h(z0))))
PROPER(g(h(c))) → c4(G(h(ok(c))), PROPER(h(c)))
PROPER(g(h(d))) → c4(G(h(ok(d))), PROPER(h(d)))
PROPER(g(c)) → c4(G(ok(c)))
K tuples:

TOP(ok(h(d))) → c11(TOP(mark(g(c))), ACTIVE(h(d)))
TOP(ok(g(z0))) → c11(TOP(mark(h(z0))), ACTIVE(g(z0)))
ACTIVE(g(z0)) → c1(H(z0))
TOP(mark(g(z0))) → c10(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(h(z0))) → c10(TOP(h(proper(z0))), PROPER(h(z0)))
Defined Rule Symbols:

active, proper, g, h, top

Defined Pair Symbols:

ACTIVE, G, H, PROPER, TOP

Compound Symbols:

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

(43) 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.

PROPER(g(c)) → c4(G(ok(c)))
We considered the (Usable) Rules:

h(ok(z0)) → ok(h(z0))
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))
And the Tuples:

ACTIVE(g(z0)) → c1(H(z0))
G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
PROPER(g(d)) → c4(G(ok(d)), PROPER(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))
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)))
PROPER(g(g(g(z0)))) → c4(G(g(g(proper(z0)))), PROPER(g(g(z0))))
PROPER(g(g(h(z0)))) → c4(G(g(h(proper(z0)))), PROPER(g(h(z0))))
PROPER(g(g(c))) → c4(G(g(ok(c))), PROPER(g(c)))
PROPER(g(g(d))) → c4(G(g(ok(d))), PROPER(g(d)))
PROPER(g(h(g(z0)))) → c4(G(h(g(proper(z0)))), PROPER(h(g(z0))))
PROPER(g(h(h(z0)))) → c4(G(h(h(proper(z0)))), PROPER(h(h(z0))))
PROPER(g(h(c))) → c4(G(h(ok(c))), PROPER(h(c)))
PROPER(g(h(d))) → c4(G(h(ok(d))), PROPER(h(d)))
PROPER(g(c)) → c4(G(ok(c)))
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)) = [1]   
POL(TOP(x1)) = [2]x1   
POL(c) = 0   
POL(c1(x1)) = x1   
POL(c10(x1, x2)) = x1 + x2   
POL(c11(x1, x2)) = x1 + x2   
POL(c4(x1)) = x1   
POL(c4(x1, x2)) = x1 + x2   
POL(c5(x1, x2)) = x1 + x2   
POL(c8(x1)) = x1   
POL(c9(x1)) = x1   
POL(d) = [2]   
POL(g(x1)) = [2] + [4]x1   
POL(h(x1)) = [4]x1   
POL(mark(x1)) = [1] + x1   
POL(ok(x1)) = x1   
POL(proper(x1)) = x1   

(44) 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(d)) → c4(G(ok(d)), PROPER(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))
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)))
PROPER(g(g(g(z0)))) → c4(G(g(g(proper(z0)))), PROPER(g(g(z0))))
PROPER(g(g(h(z0)))) → c4(G(g(h(proper(z0)))), PROPER(g(h(z0))))
PROPER(g(g(c))) → c4(G(g(ok(c))), PROPER(g(c)))
PROPER(g(g(d))) → c4(G(g(ok(d))), PROPER(g(d)))
PROPER(g(h(g(z0)))) → c4(G(h(g(proper(z0)))), PROPER(h(g(z0))))
PROPER(g(h(h(z0)))) → c4(G(h(h(proper(z0)))), PROPER(h(h(z0))))
PROPER(g(h(c))) → c4(G(h(ok(c))), PROPER(h(c)))
PROPER(g(h(d))) → c4(G(h(ok(d))), PROPER(h(d)))
PROPER(g(c)) → c4(G(ok(c)))
S tuples:

G(ok(z0)) → c8(G(z0))
H(ok(z0)) → c9(H(z0))
PROPER(g(d)) → c4(G(ok(d)), PROPER(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))
PROPER(g(g(g(z0)))) → c4(G(g(g(proper(z0)))), PROPER(g(g(z0))))
PROPER(g(g(h(z0)))) → c4(G(g(h(proper(z0)))), PROPER(g(h(z0))))
PROPER(g(g(c))) → c4(G(g(ok(c))), PROPER(g(c)))
PROPER(g(g(d))) → c4(G(g(ok(d))), PROPER(g(d)))
PROPER(g(h(g(z0)))) → c4(G(h(g(proper(z0)))), PROPER(h(g(z0))))
PROPER(g(h(h(z0)))) → c4(G(h(h(proper(z0)))), PROPER(h(h(z0))))
PROPER(g(h(c))) → c4(G(h(ok(c))), PROPER(h(c)))
PROPER(g(h(d))) → c4(G(h(ok(d))), PROPER(h(d)))
K tuples:

TOP(ok(h(d))) → c11(TOP(mark(g(c))), ACTIVE(h(d)))
TOP(ok(g(z0))) → c11(TOP(mark(h(z0))), ACTIVE(g(z0)))
ACTIVE(g(z0)) → c1(H(z0))
TOP(mark(g(z0))) → c10(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(h(z0))) → c10(TOP(h(proper(z0))), PROPER(h(z0)))
PROPER(g(c)) → c4(G(ok(c)))
Defined Rule Symbols:

active, proper, g, h, top

Defined Pair Symbols:

ACTIVE, G, H, PROPER, TOP

Compound Symbols:

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

(45) CpxTrsMatchBoundsProof (EQUIVALENT transformation)

A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 4.
The certificate found is represented by the following graph.
Start state: 2796
Accept states: [2797, 2798, 2799, 2800, 2801]
Transitions:
2796→2797[active_1|0]
2796→2798[proper_1|0]
2796→2799[g_1|0]
2796→2800[h_1|0]
2796→2801[top_1|0]
2796→2796[mark_1|0, c|0, d|0, ok_1|0]
2796→2802[d|1]
2796→2803[c|1]
2796→2804[d|1]
2796→2805[g_1|1]
2796→2806[h_1|1]
2796→2807[active_1|1]
2796→2808[proper_1|1]
2796→2811[d|2]
2796→2812[d|2]
2796→2815[d|3]
2802→2797[mark_1|1]
2802→2807[mark_1|1]
2802→2809[proper_1|2]
2803→2798[ok_1|1]
2803→2808[ok_1|1]
2803→2810[active_1|2]
2804→2798[ok_1|1]
2804→2808[ok_1|1]
2804→2810[active_1|2]
2805→2799[ok_1|1]
2805→2805[ok_1|1]
2806→2800[ok_1|1]
2806→2806[ok_1|1]
2807→2801[top_1|1]
2808→2801[top_1|1]
2809→2801[top_1|2]
2810→2801[top_1|2]
2811→2809[ok_1|2]
2811→2813[active_1|3]
2812→2810[mark_1|2]
2812→2814[proper_1|3]
2813→2801[top_1|3]
2814→2801[top_1|3]
2815→2814[ok_1|3]
2815→2816[active_1|4]
2816→2801[top_1|4]

(46) BOUNDS(O(1), O(n^1))