Runtime Complexity (innermost) proof of /tmp/tmpArjen5/ExConc_Zan97

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 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳4 CdtProblem

    ↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 467 ms)

    ↳6 CdtProblem

    ↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 527 ms)

    ↳8 CdtProblem

    ↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 557 ms)

    ↳10 CdtProblem

    ↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))), 1201 ms)

    ↳12 CdtProblem

    ↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))), 491 ms)

    ↳14 CdtProblem

    ↳15 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))), 396 ms)

    ↳16 CdtProblem

    ↳17 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))), 418 ms)

    ↳18 CdtProblem

    ↳19 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))), 1114 ms)

    ↳20 CdtProblem

    ↳21 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))), 1662 ms)

    ↳22 CdtProblem

    ↳23 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳24 CdtProblem

    ↳25 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳26 CdtProblem

    ↳27 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳28 CdtProblem

    ↳29 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳30 CdtProblem

    ↳31 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳32 CdtProblem

    ↳33 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳34 CdtProblem

    ↳35 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 3 ms)

    ↳36 CdtProblem

    ↳37 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳38 CdtProblem

    ↳39 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳40 CdtProblem

    ↳41 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳42 CdtProblem

    ↳43 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳44 CdtProblem

    ↳45 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳46 CdtProblem

↳47 CpxTrsMatchBoundsProof (⇔, 0 ms)

↳48 BOUNDS(O(1), O(n^1))

(0) Obligation:

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

active(f(X)) → mark(g(h(f(X))))
active(f(X)) → f(active(X))
active(h(X)) → h(active(X))
f(mark(X)) → mark(f(X))
h(mark(X)) → mark(h(X))
proper(f(X)) → f(proper(X))
proper(g(X)) → g(proper(X))
proper(h(X)) → h(proper(X))
f(ok(X)) → ok(f(X))
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(f(z0)) → mark(g(h(f(z0))))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
proper(f(z0)) → f(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(f(z0)) → c(G(h(f(z0))), H(f(z0)), F(z0))
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(f(z0)) → c(G(h(f(z0))), H(f(z0)), F(z0))
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, G, TOP

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12

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

Removed 2 trailing tuple parts

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(z0))))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
proper(f(z0)) → f(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))

S tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))

K tuples:none
Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, G, TOP

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c

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

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

G(ok(z0)) → c10(G(z0))

We considered the (Usable) Rules:

active(f(z0)) → mark(g(h(f(z0))))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
proper(f(z0)) → f(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))

And the Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))

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

POL(ACTIVE(x₁)) = 0
POL(F(x₁)) = 0
POL(G(x₁)) = [4]x₁
POL(H(x₁)) = 0
POL(PROPER(x₁)) = 0
POL(TOP(x₁)) = 0
POL(active(x₁)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁, x₂)) = x₁ + x₂
POL(c10(x₁)) = x₁
POL(c11(x₁, x₂)) = x₁ + x₂
POL(c12(x₁, x₂)) = x₁ + x₂
POL(c2(x₁, x₂)) = x₁ + x₂
POL(c3(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁, x₂)) = x₁ + x₂
POL(c8(x₁, x₂)) = x₁ + x₂
POL(c9(x₁, x₂)) = x₁ + x₂
POL(f(x₁)) = [2]x₁
POL(g(x₁)) = x₁
POL(h(x₁)) = [2]x₁
POL(mark(x₁)) = 0
POL(ok(x₁)) = [1] + x₁
POL(proper(x₁)) = 0

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(z0))))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
proper(f(z0)) → f(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))

S tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))

K tuples:

G(ok(z0)) → c10(G(z0))

Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, G, TOP

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c

(7) 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(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))

We considered the (Usable) Rules:

active(f(z0)) → mark(g(h(f(z0))))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
proper(f(z0)) → f(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))

And the Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))

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

POL(ACTIVE(x₁)) = [1]
POL(F(x₁)) = 0
POL(G(x₁)) = 0
POL(H(x₁)) = 0
POL(PROPER(x₁)) = 0
POL(TOP(x₁)) = [2]x₁
POL(active(x₁)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁, x₂)) = x₁ + x₂
POL(c10(x₁)) = x₁
POL(c11(x₁, x₂)) = x₁ + x₂
POL(c12(x₁, x₂)) = x₁ + x₂
POL(c2(x₁, x₂)) = x₁ + x₂
POL(c3(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁, x₂)) = x₁ + x₂
POL(c8(x₁, x₂)) = x₁ + x₂
POL(c9(x₁, x₂)) = x₁ + x₂
POL(f(x₁)) = [4]x₁
POL(g(x₁)) = [2]x₁
POL(h(x₁)) = [2]x₁
POL(mark(x₁)) = 0
POL(ok(x₁)) = [2]
POL(proper(x₁)) = 0

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(z0))))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
proper(f(z0)) → f(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))

S tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))

K tuples:

G(ok(z0)) → c10(G(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))

Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, G, TOP

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c

(9) 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(z0)) → c11(TOP(proper(z0)), PROPER(z0))

We considered the (Usable) Rules:

active(f(z0)) → mark(g(h(f(z0))))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
proper(f(z0)) → f(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))

And the Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))

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

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

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(z0))))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
proper(f(z0)) → f(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))

S tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))

K tuples:

G(ok(z0)) → c10(G(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, G, TOP

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c

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

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

F(ok(z0)) → c4(F(z0))

We considered the (Usable) Rules:

active(f(z0)) → mark(g(h(f(z0))))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
proper(f(z0)) → f(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))

And the Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))

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

POL(ACTIVE(x₁)) = x₁
POL(F(x₁)) = [2]x₁²
POL(G(x₁)) = 0
POL(H(x₁)) = 0
POL(PROPER(x₁)) = 0
POL(TOP(x₁)) = [2]x₁ + x₁²
POL(active(x₁)) = x₁
POL(c(x₁)) = x₁
POL(c1(x₁, x₂)) = x₁ + x₂
POL(c10(x₁)) = x₁
POL(c11(x₁, x₂)) = x₁ + x₂
POL(c12(x₁, x₂)) = x₁ + x₂
POL(c2(x₁, x₂)) = x₁ + x₂
POL(c3(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁, x₂)) = x₁ + x₂
POL(c8(x₁, x₂)) = x₁ + x₂
POL(c9(x₁, x₂)) = x₁ + x₂
POL(f(x₁)) = x₁ + [2]x₁²
POL(g(x₁)) = x₁
POL(h(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(ok(x₁)) = [3] + x₁
POL(proper(x₁)) = 0

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(z0))))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
proper(f(z0)) → f(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))

S tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))

K tuples:

G(ok(z0)) → c10(G(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
F(ok(z0)) → c4(F(z0))

Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, G, TOP

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c

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

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

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))

We considered the (Usable) Rules:

active(f(z0)) → mark(g(h(f(z0))))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
proper(f(z0)) → f(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))

And the Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))

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

POL(ACTIVE(x₁)) = [1] + [2]x₁
POL(F(x₁)) = 0
POL(G(x₁)) = 0
POL(H(x₁)) = 0
POL(PROPER(x₁)) = 0
POL(TOP(x₁)) = x₁²
POL(active(x₁)) = x₁
POL(c(x₁)) = x₁
POL(c1(x₁, x₂)) = x₁ + x₂
POL(c10(x₁)) = x₁
POL(c11(x₁, x₂)) = x₁ + x₂
POL(c12(x₁, x₂)) = x₁ + x₂
POL(c2(x₁, x₂)) = x₁ + x₂
POL(c3(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁, x₂)) = x₁ + x₂
POL(c8(x₁, x₂)) = x₁ + x₂
POL(c9(x₁, x₂)) = x₁ + x₂
POL(f(x₁)) = [1] + x₁
POL(g(x₁)) = x₁
POL(h(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(ok(x₁)) = [1] + x₁
POL(proper(x₁)) = x₁

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(z0))))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
proper(f(z0)) → f(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))

S tuples:

ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))

K tuples:

G(ok(z0)) → c10(G(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
F(ok(z0)) → c4(F(z0))
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))

Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, G, TOP

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c

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

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

PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))

We considered the (Usable) Rules:

active(f(z0)) → mark(g(h(f(z0))))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
proper(f(z0)) → f(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))

And the Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))

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

POL(ACTIVE(x₁)) = x₁
POL(F(x₁)) = 0
POL(G(x₁)) = 0
POL(H(x₁)) = 0
POL(PROPER(x₁)) = [3] + x₁
POL(TOP(x₁)) = x₁ + [2]x₁²
POL(active(x₁)) = [1] + x₁
POL(c(x₁)) = x₁
POL(c1(x₁, x₂)) = x₁ + x₂
POL(c10(x₁)) = x₁
POL(c11(x₁, x₂)) = x₁ + x₂
POL(c12(x₁, x₂)) = x₁ + x₂
POL(c2(x₁, x₂)) = x₁ + x₂
POL(c3(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁, x₂)) = x₁ + x₂
POL(c8(x₁, x₂)) = x₁ + x₂
POL(c9(x₁, x₂)) = x₁ + x₂
POL(f(x₁)) = [2] + x₁
POL(g(x₁)) = x₁
POL(h(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = [3] + x₁
POL(proper(x₁)) = x₁

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(z0))))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
proper(f(z0)) → f(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))

S tuples:

ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))

K tuples:

G(ok(z0)) → c10(G(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
F(ok(z0)) → c4(F(z0))
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, G, TOP

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c

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

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

PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))

We considered the (Usable) Rules:

active(f(z0)) → mark(g(h(f(z0))))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
proper(f(z0)) → f(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))

And the Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))

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

POL(ACTIVE(x₁)) = 0
POL(F(x₁)) = 0
POL(G(x₁)) = 0
POL(H(x₁)) = 0
POL(PROPER(x₁)) = [2]x₁
POL(TOP(x₁)) = x₁²
POL(active(x₁)) = [3] + x₁
POL(c(x₁)) = x₁
POL(c1(x₁, x₂)) = x₁ + x₂
POL(c10(x₁)) = x₁
POL(c11(x₁, x₂)) = x₁ + x₂
POL(c12(x₁, x₂)) = x₁ + x₂
POL(c2(x₁, x₂)) = x₁ + x₂
POL(c3(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁, x₂)) = x₁ + x₂
POL(c8(x₁, x₂)) = x₁ + x₂
POL(c9(x₁, x₂)) = x₁ + x₂
POL(f(x₁)) = x₁
POL(g(x₁)) = x₁
POL(h(x₁)) = [1] + x₁
POL(mark(x₁)) = [2] + x₁
POL(ok(x₁)) = [3] + x₁
POL(proper(x₁)) = [1] + x₁

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(z0))))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
proper(f(z0)) → f(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))

S tuples:

ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))

K tuples:

G(ok(z0)) → c10(G(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
F(ok(z0)) → c4(F(z0))
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, G, TOP

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c

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

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

ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))

We considered the (Usable) Rules:

active(f(z0)) → mark(g(h(f(z0))))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
proper(f(z0)) → f(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))

And the Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))

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

POL(ACTIVE(x₁)) = x₁
POL(F(x₁)) = 0
POL(G(x₁)) = 0
POL(H(x₁)) = 0
POL(PROPER(x₁)) = 0
POL(TOP(x₁)) = [2]x₁²
POL(active(x₁)) = [2] + x₁
POL(c(x₁)) = x₁
POL(c1(x₁, x₂)) = x₁ + x₂
POL(c10(x₁)) = x₁
POL(c11(x₁, x₂)) = x₁ + x₂
POL(c12(x₁, x₂)) = x₁ + x₂
POL(c2(x₁, x₂)) = x₁ + x₂
POL(c3(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁, x₂)) = x₁ + x₂
POL(c8(x₁, x₂)) = x₁ + x₂
POL(c9(x₁, x₂)) = x₁ + x₂
POL(f(x₁)) = x₁
POL(g(x₁)) = x₁
POL(h(x₁)) = [1] + x₁
POL(mark(x₁)) = x₁
POL(ok(x₁)) = [3] + x₁
POL(proper(x₁)) = x₁

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(z0))))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
proper(f(z0)) → f(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))

S tuples:

F(mark(z0)) → c3(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))

K tuples:

G(ok(z0)) → c10(G(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
F(ok(z0)) → c4(F(z0))
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))

Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, G, TOP

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c

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

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

PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))

We considered the (Usable) Rules:

active(f(z0)) → mark(g(h(f(z0))))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
proper(f(z0)) → f(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))

And the Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))

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

POL(ACTIVE(x₁)) = 0
POL(F(x₁)) = 0
POL(G(x₁)) = [3]
POL(H(x₁)) = 0
POL(PROPER(x₁)) = [2] + [2]x₁
POL(TOP(x₁)) = [2]x₁ + [2]x₁²
POL(active(x₁)) = [3] + x₁
POL(c(x₁)) = x₁
POL(c1(x₁, x₂)) = x₁ + x₂
POL(c10(x₁)) = x₁
POL(c11(x₁, x₂)) = x₁ + x₂
POL(c12(x₁, x₂)) = x₁ + x₂
POL(c2(x₁, x₂)) = x₁ + x₂
POL(c3(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁, x₂)) = x₁ + x₂
POL(c8(x₁, x₂)) = x₁ + x₂
POL(c9(x₁, x₂)) = x₁ + x₂
POL(f(x₁)) = x₁
POL(g(x₁)) = [2] + x₁
POL(h(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = [3] + x₁
POL(proper(x₁)) = x₁

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(z0))))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
proper(f(z0)) → f(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))

S tuples:

F(mark(z0)) → c3(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))

K tuples:

G(ok(z0)) → c10(G(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
F(ok(z0)) → c4(F(z0))
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, G, TOP

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c

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

Use narrowing to replace ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0)) by

ACTIVE(f(f(z0))) → c1(F(mark(g(h(f(z0))))), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(h(z0))) → c1(F(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(f(x0)) → c1

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(z0))))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
proper(f(z0)) → f(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))
ACTIVE(f(f(z0))) → c1(F(mark(g(h(f(z0))))), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(h(z0))) → c1(F(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(f(x0)) → c1

S tuples:

F(mark(z0)) → c3(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))

K tuples:

G(ok(z0)) → c10(G(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
F(ok(z0)) → c4(F(z0))
ACTIVE(f(z0)) → c1(F(active(z0)), ACTIVE(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, G, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c, c1, c1

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

Removed 1 trailing nodes:

ACTIVE(f(x0)) → c1

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(z0))))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
proper(f(z0)) → f(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))
ACTIVE(f(f(z0))) → c1(F(mark(g(h(f(z0))))), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(h(z0))) → c1(F(h(active(z0))), ACTIVE(h(z0)))

S tuples:

F(mark(z0)) → c3(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))

K tuples:

G(ok(z0)) → c10(G(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
F(ok(z0)) → c4(F(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

ACTIVE, F, H, PROPER, G, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c, c1

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

Use narrowing to replace ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0)) by

ACTIVE(h(f(z0))) → c2(H(mark(g(h(f(z0))))), ACTIVE(f(z0)))
ACTIVE(h(f(z0))) → c2(H(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(h(h(z0))) → c2(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(x0)) → c2

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(z0))))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
proper(f(z0)) → f(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))
ACTIVE(f(f(z0))) → c1(F(mark(g(h(f(z0))))), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(h(z0))) → c1(F(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0))) → c2(H(mark(g(h(f(z0))))), ACTIVE(f(z0)))
ACTIVE(h(f(z0))) → c2(H(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(h(h(z0))) → c2(H(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(x0)) → c2

S tuples:

F(mark(z0)) → c3(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))

K tuples:

G(ok(z0)) → c10(G(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
F(ok(z0)) → c4(F(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
ACTIVE(h(z0)) → c2(H(active(z0)), ACTIVE(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

F, H, PROPER, G, TOP, ACTIVE

Compound Symbols:

c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c, c1, c2, c2

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

Removed 1 trailing nodes:

ACTIVE(h(x0)) → c2

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(z0))))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
proper(f(z0)) → f(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))
ACTIVE(f(f(z0))) → c1(F(mark(g(h(f(z0))))), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(h(z0))) → c1(F(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0))) → c2(H(mark(g(h(f(z0))))), ACTIVE(f(z0)))
ACTIVE(h(f(z0))) → c2(H(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(h(h(z0))) → c2(H(h(active(z0))), ACTIVE(h(z0)))

S tuples:

F(mark(z0)) → c3(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))

K tuples:

G(ok(z0)) → c10(G(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
F(ok(z0)) → c4(F(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

F, H, PROPER, G, TOP, ACTIVE

Compound Symbols:

c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c, c1, c2

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

Use narrowing to replace PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0)) by

PROPER(f(f(z0))) → c7(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c7(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(h(z0))) → c7(F(h(proper(z0))), PROPER(h(z0)))
PROPER(f(x0)) → c7

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(z0))))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
proper(f(z0)) → f(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))
ACTIVE(f(f(z0))) → c1(F(mark(g(h(f(z0))))), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(h(z0))) → c1(F(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0))) → c2(H(mark(g(h(f(z0))))), ACTIVE(f(z0)))
ACTIVE(h(f(z0))) → c2(H(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(h(h(z0))) → c2(H(h(active(z0))), ACTIVE(h(z0)))
PROPER(f(f(z0))) → c7(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c7(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(h(z0))) → c7(F(h(proper(z0))), PROPER(h(z0)))
PROPER(f(x0)) → c7

S tuples:

F(mark(z0)) → c3(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))

K tuples:

G(ok(z0)) → c10(G(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
F(ok(z0)) → c4(F(z0))
PROPER(f(z0)) → c7(F(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

F, H, PROPER, G, TOP, ACTIVE

Compound Symbols:

c3, c4, c5, c6, c8, c9, c10, c11, c12, c, c1, c2, c7, c7

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

Removed 1 trailing nodes:

PROPER(f(x0)) → c7

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(z0))))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
proper(f(z0)) → f(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))
ACTIVE(f(f(z0))) → c1(F(mark(g(h(f(z0))))), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(h(z0))) → c1(F(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0))) → c2(H(mark(g(h(f(z0))))), ACTIVE(f(z0)))
ACTIVE(h(f(z0))) → c2(H(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(h(h(z0))) → c2(H(h(active(z0))), ACTIVE(h(z0)))
PROPER(f(f(z0))) → c7(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c7(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(h(z0))) → c7(F(h(proper(z0))), PROPER(h(z0)))

S tuples:

F(mark(z0)) → c3(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))

K tuples:

G(ok(z0)) → c10(G(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
F(ok(z0)) → c4(F(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

F, H, PROPER, G, TOP, ACTIVE

Compound Symbols:

c3, c4, c5, c6, c8, c9, c10, c11, c12, c, c1, c2, c7

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

Use narrowing to replace PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0)) by

PROPER(g(f(z0))) → c8(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c8(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(x0)) → c8

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(z0))))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
proper(f(z0)) → f(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))
ACTIVE(f(f(z0))) → c1(F(mark(g(h(f(z0))))), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(h(z0))) → c1(F(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0))) → c2(H(mark(g(h(f(z0))))), ACTIVE(f(z0)))
ACTIVE(h(f(z0))) → c2(H(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(h(h(z0))) → c2(H(h(active(z0))), ACTIVE(h(z0)))
PROPER(f(f(z0))) → c7(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c7(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(h(z0))) → c7(F(h(proper(z0))), PROPER(h(z0)))
PROPER(g(f(z0))) → c8(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c8(G(h(proper(z0))), PROPER(h(z0)))
PROPER(g(x0)) → c8

S tuples:

F(mark(z0)) → c3(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))

K tuples:

G(ok(z0)) → c10(G(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
F(ok(z0)) → c4(F(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

F, H, PROPER, G, TOP, ACTIVE

Compound Symbols:

c3, c4, c5, c6, c9, c10, c11, c12, c, c1, c2, c7, c8, c8

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

Removed 1 trailing nodes:

PROPER(g(x0)) → c8

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(z0))))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
proper(f(z0)) → f(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))
ACTIVE(f(f(z0))) → c1(F(mark(g(h(f(z0))))), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(h(z0))) → c1(F(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0))) → c2(H(mark(g(h(f(z0))))), ACTIVE(f(z0)))
ACTIVE(h(f(z0))) → c2(H(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(h(h(z0))) → c2(H(h(active(z0))), ACTIVE(h(z0)))
PROPER(f(f(z0))) → c7(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c7(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(h(z0))) → c7(F(h(proper(z0))), PROPER(h(z0)))
PROPER(g(f(z0))) → c8(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c8(G(h(proper(z0))), PROPER(h(z0)))

S tuples:

F(mark(z0)) → c3(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))

K tuples:

G(ok(z0)) → c10(G(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
F(ok(z0)) → c4(F(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

F, H, PROPER, G, TOP, ACTIVE

Compound Symbols:

c3, c4, c5, c6, c9, c10, c11, c12, c, c1, c2, c7, c8

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

Use narrowing to replace PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0)) by

PROPER(h(f(z0))) → c9(H(f(proper(z0))), PROPER(f(z0)))
PROPER(h(g(z0))) → c9(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c9(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(x0)) → c9

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(z0))))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
proper(f(z0)) → f(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))
ACTIVE(f(f(z0))) → c1(F(mark(g(h(f(z0))))), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(h(z0))) → c1(F(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0))) → c2(H(mark(g(h(f(z0))))), ACTIVE(f(z0)))
ACTIVE(h(f(z0))) → c2(H(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(h(h(z0))) → c2(H(h(active(z0))), ACTIVE(h(z0)))
PROPER(f(f(z0))) → c7(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c7(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(h(z0))) → c7(F(h(proper(z0))), PROPER(h(z0)))
PROPER(g(f(z0))) → c8(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c8(G(h(proper(z0))), PROPER(h(z0)))
PROPER(h(f(z0))) → c9(H(f(proper(z0))), PROPER(f(z0)))
PROPER(h(g(z0))) → c9(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c9(H(h(proper(z0))), PROPER(h(z0)))
PROPER(h(x0)) → c9

S tuples:

F(mark(z0)) → c3(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))

K tuples:

G(ok(z0)) → c10(G(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
F(ok(z0)) → c4(F(z0))
PROPER(h(z0)) → c9(H(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

F, H, G, TOP, ACTIVE, PROPER

Compound Symbols:

c3, c4, c5, c6, c10, c11, c12, c, c1, c2, c7, c8, c9, c9

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

Removed 1 trailing nodes:

PROPER(h(x0)) → c9

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(z0))))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
proper(f(z0)) → f(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
G(ok(z0)) → c10(G(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))
ACTIVE(f(f(z0))) → c1(F(mark(g(h(f(z0))))), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(h(z0))) → c1(F(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0))) → c2(H(mark(g(h(f(z0))))), ACTIVE(f(z0)))
ACTIVE(h(f(z0))) → c2(H(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(h(h(z0))) → c2(H(h(active(z0))), ACTIVE(h(z0)))
PROPER(f(f(z0))) → c7(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c7(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(h(z0))) → c7(F(h(proper(z0))), PROPER(h(z0)))
PROPER(g(f(z0))) → c8(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c8(G(h(proper(z0))), PROPER(h(z0)))
PROPER(h(f(z0))) → c9(H(f(proper(z0))), PROPER(f(z0)))
PROPER(h(g(z0))) → c9(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c9(H(h(proper(z0))), PROPER(h(z0)))

S tuples:

F(mark(z0)) → c3(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))

K tuples:

G(ok(z0)) → c10(G(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
F(ok(z0)) → c4(F(z0))

Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

F, H, G, TOP, ACTIVE, PROPER

Compound Symbols:

c3, c4, c5, c6, c10, c11, c12, c, c1, c2, c7, c8, c9

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

Use narrowing to replace TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(f(z0))) → c11(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(g(z0))) → c11(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(h(z0))) → c11(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(x0)) → c11

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(z0))))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
proper(f(z0)) → f(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
G(ok(z0)) → c10(G(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))
ACTIVE(f(f(z0))) → c1(F(mark(g(h(f(z0))))), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(h(z0))) → c1(F(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0))) → c2(H(mark(g(h(f(z0))))), ACTIVE(f(z0)))
ACTIVE(h(f(z0))) → c2(H(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(h(h(z0))) → c2(H(h(active(z0))), ACTIVE(h(z0)))
PROPER(f(f(z0))) → c7(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c7(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(h(z0))) → c7(F(h(proper(z0))), PROPER(h(z0)))
PROPER(g(f(z0))) → c8(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c8(G(h(proper(z0))), PROPER(h(z0)))
PROPER(h(f(z0))) → c9(H(f(proper(z0))), PROPER(f(z0)))
PROPER(h(g(z0))) → c9(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c9(H(h(proper(z0))), PROPER(h(z0)))
TOP(mark(f(z0))) → c11(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(g(z0))) → c11(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(h(z0))) → c11(TOP(h(proper(z0))), PROPER(h(z0)))
TOP(mark(x0)) → c11

S tuples:

F(mark(z0)) → c3(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))

K tuples:

G(ok(z0)) → c10(G(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))
TOP(mark(z0)) → c11(TOP(proper(z0)), PROPER(z0))
F(ok(z0)) → c4(F(z0))

Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

F, H, G, TOP, ACTIVE, PROPER

Compound Symbols:

c3, c4, c5, c6, c10, c12, c, c1, c2, c7, c8, c9, c11, c11

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

Removed 1 trailing nodes:

TOP(mark(x0)) → c11

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(g(h(f(z0))))
active(f(z0)) → f(active(z0))
active(h(z0)) → h(active(z0))
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
h(mark(z0)) → mark(h(z0))
h(ok(z0)) → ok(h(z0))
proper(f(z0)) → f(proper(z0))
proper(g(z0)) → g(proper(z0))
proper(h(z0)) → h(proper(z0))
g(ok(z0)) → ok(g(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

F(mark(z0)) → c3(F(z0))
F(ok(z0)) → c4(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))
G(ok(z0)) → c10(G(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))
ACTIVE(f(f(z0))) → c1(F(mark(g(h(f(z0))))), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c1(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(h(z0))) → c1(F(h(active(z0))), ACTIVE(h(z0)))
ACTIVE(h(f(z0))) → c2(H(mark(g(h(f(z0))))), ACTIVE(f(z0)))
ACTIVE(h(f(z0))) → c2(H(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(h(h(z0))) → c2(H(h(active(z0))), ACTIVE(h(z0)))
PROPER(f(f(z0))) → c7(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(g(z0))) → c7(F(g(proper(z0))), PROPER(g(z0)))
PROPER(f(h(z0))) → c7(F(h(proper(z0))), PROPER(h(z0)))
PROPER(g(f(z0))) → c8(G(f(proper(z0))), PROPER(f(z0)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(h(z0))) → c8(G(h(proper(z0))), PROPER(h(z0)))
PROPER(h(f(z0))) → c9(H(f(proper(z0))), PROPER(f(z0)))
PROPER(h(g(z0))) → c9(H(g(proper(z0))), PROPER(g(z0)))
PROPER(h(h(z0))) → c9(H(h(proper(z0))), PROPER(h(z0)))
TOP(mark(f(z0))) → c11(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(g(z0))) → c11(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(h(z0))) → c11(TOP(h(proper(z0))), PROPER(h(z0)))

S tuples:

F(mark(z0)) → c3(F(z0))
H(mark(z0)) → c5(H(z0))
H(ok(z0)) → c6(H(z0))

K tuples:

G(ok(z0)) → c10(G(z0))
TOP(ok(z0)) → c12(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c(F(z0))
F(ok(z0)) → c4(F(z0))

Defined Rule Symbols:

active, f, h, proper, g, top

Defined Pair Symbols:

F, H, G, TOP, ACTIVE, PROPER

Compound Symbols:

c3, c4, c5, c6, c10, c12, c, c1, c2, c7, c8, c9, c11

(47) 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 1.
The certificate found is represented by the following graph.

Start state: 3288
Accept states: [3289, 3290, 3291, 3292, 3293, 3294]
Transitions:
3288→3289[active_1|0]
3288→3290[f_1|0]
3288→3291[h_1|0]
3288→3292[proper_1|0]
3288→3293[g_1|0]
3288→3294[top_1|0]
3288→3288[mark_1|0, ok_1|0]
3288→3295[f_1|1]
3288→3296[h_1|1]
3288→3297[proper_1|1]
3288→3298[f_1|1]
3288→3299[h_1|1]
3288→3300[g_1|1]
3288→3301[active_1|1]
3295→3290[mark_1|1]
3295→3295[mark_1|1]
3295→3298[mark_1|1]
3296→3291[mark_1|1]
3296→3296[mark_1|1]
3296→3299[mark_1|1]
3297→3294[top_1|1]
3298→3290[ok_1|1]
3298→3295[ok_1|1]
3298→3298[ok_1|1]
3299→3291[ok_1|1]
3299→3296[ok_1|1]
3299→3299[ok_1|1]
3300→3293[ok_1|1]
3300→3300[ok_1|1]
3301→3294[top_1|1]

(0) Obligation:

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

(2) Obligation:

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

(4) Obligation:

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

(6) Obligation:

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

(8) Obligation:

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

(10) Obligation:

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

(12) Obligation:

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

(14) Obligation:

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

(16) Obligation:

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

(18) Obligation:

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

(20) Obligation:

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

(22) Obligation:

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

(24) Obligation:

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

(26) Obligation:

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

(28) Obligation:

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

(30) Obligation:

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

(32) Obligation:

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

(34) Obligation:

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

(36) Obligation:

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

(38) Obligation:

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

(40) Obligation:

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

(42) Obligation:

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

(44) Obligation:

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

(46) Obligation:

(47) CpxTrsMatchBoundsProof (EQUIVALENT transformation)

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