The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

(0) Obligation:

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

active(f(g(X), Y)) → mark(f(X, f(g(X), Y)))
active(f(X1, X2)) → f(active(X1), X2)
active(g(X)) → g(active(X))
f(mark(X1), X2) → mark(f(X1, X2))
g(mark(X)) → mark(g(X))
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(g(X)) → g(proper(X))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
g(ok(X)) → ok(g(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(g(z0), z1)) → mark(f(z0, f(g(z0), z1)))
active(f(z0, z1)) → f(active(z0), z1)
active(g(z0)) → g(active(z0))
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
g(mark(z0)) → mark(g(z0))
g(ok(z0)) → ok(g(z0))
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(g(z0)) → g(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(f(g(z0), z1)) → c(F(z0, f(g(z0), z1)), F(g(z0), z1), G(z0))
ACTIVE(f(z0, z1)) → c1(F(active(z0), z1), ACTIVE(z0))
ACTIVE(g(z0)) → c2(G(active(z0)), ACTIVE(z0))
F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(f(z0, z1)) → c7(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(f(g(z0), z1)) → c(F(z0, f(g(z0), z1)), F(g(z0), z1), G(z0))
ACTIVE(f(z0, z1)) → c1(F(active(z0), z1), ACTIVE(z0))
ACTIVE(g(z0)) → c2(G(active(z0)), ACTIVE(z0))
F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(f(z0, z1)) → c7(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, f, g, proper, top

Defined Pair Symbols:

ACTIVE, F, G, PROPER, TOP

Compound Symbols:

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

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

Removed 2 trailing tuple parts

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ACTIVE(f(z0, z1)) → c1(F(active(z0), z1), ACTIVE(z0))
ACTIVE(g(z0)) → c2(G(active(z0)), ACTIVE(z0))
F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(f(z0, z1)) → c7(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(g(z0), z1)) → c(G(z0))

S tuples:

ACTIVE(f(z0, z1)) → c1(F(active(z0), z1), ACTIVE(z0))
ACTIVE(g(z0)) → c2(G(active(z0)), ACTIVE(z0))
F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(f(z0, z1)) → c7(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(g(z0), z1)) → c(G(z0))

K tuples:none
Defined Rule Symbols:

active, f, g, proper, top

Defined Pair Symbols:

ACTIVE, F, G, PROPER, TOP

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, 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.

TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(g(z0), z1)) → c(G(z0))

We considered the (Usable) Rules:

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

And the Tuples:

ACTIVE(f(z0, z1)) → c1(F(active(z0), z1), ACTIVE(z0))
ACTIVE(g(z0)) → c2(G(active(z0)), ACTIVE(z0))
F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(f(z0, z1)) → c7(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(g(z0), z1)) → c(G(z0))

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

POL(ACTIVE(x₁)) = [2]   
POL(F(x₁, x₂)) = 0   
POL(G(x₁)) = 0   
POL(PROPER(x₁)) = 0   
POL(TOP(x₁)) = [4]x₁   
POL(active(x₁)) = x₁   
POL(c(x₁)) = x₁   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c10(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, x₃)) = x₁ + x₂ + x₃   
POL(c8(x₁, x₂)) = x₁ + x₂   
POL(c9(x₁, x₂)) = x₁ + x₂   
POL(f(x₁, x₂)) = [2]x₁   
POL(g(x₁)) = [4]x₁   
POL(mark(x₁)) = x₁   
POL(ok(x₁)) = [2] + x₁   
POL(proper(x₁)) = 0   

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ACTIVE(f(z0, z1)) → c1(F(active(z0), z1), ACTIVE(z0))
ACTIVE(g(z0)) → c2(G(active(z0)), ACTIVE(z0))
F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(f(z0, z1)) → c7(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(g(z0), z1)) → c(G(z0))

S tuples:

ACTIVE(f(z0, z1)) → c1(F(active(z0), z1), ACTIVE(z0))
ACTIVE(g(z0)) → c2(G(active(z0)), ACTIVE(z0))
F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(f(z0, z1)) → c7(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))

K tuples:

TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(g(z0), z1)) → c(G(z0))

Defined Rule Symbols:

active, f, g, proper, top

Defined Pair Symbols:

ACTIVE, F, G, PROPER, TOP

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, 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(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))

We considered the (Usable) Rules:

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

And the Tuples:

ACTIVE(f(z0, z1)) → c1(F(active(z0), z1), ACTIVE(z0))
ACTIVE(g(z0)) → c2(G(active(z0)), ACTIVE(z0))
F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(f(z0, z1)) → c7(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(g(z0), z1)) → c(G(z0))

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

POL(ACTIVE(x₁)) = 0   
POL(F(x₁, x₂)) = 0   
POL(G(x₁)) = 0   
POL(PROPER(x₁)) = 0   
POL(TOP(x₁)) = [2]x₁   
POL(active(x₁)) = [4]   
POL(c(x₁)) = x₁   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c10(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, x₃)) = x₁ + x₂ + x₃   
POL(c8(x₁, x₂)) = x₁ + x₂   
POL(c9(x₁, x₂)) = x₁ + x₂   
POL(f(x₁, x₂)) = x₁   
POL(g(x₁)) = x₁   
POL(mark(x₁)) = [4]   
POL(ok(x₁)) = [4] + x₁   
POL(proper(x₁)) = 0   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ACTIVE(f(z0, z1)) → c1(F(active(z0), z1), ACTIVE(z0))
ACTIVE(g(z0)) → c2(G(active(z0)), ACTIVE(z0))
F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(f(z0, z1)) → c7(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(g(z0), z1)) → c(G(z0))

S tuples:

ACTIVE(f(z0, z1)) → c1(F(active(z0), z1), ACTIVE(z0))
ACTIVE(g(z0)) → c2(G(active(z0)), ACTIVE(z0))
F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(f(z0, z1)) → c7(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))

K tuples:

TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(g(z0), z1)) → c(G(z0))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, f, g, proper, top

Defined Pair Symbols:

ACTIVE, F, G, PROPER, TOP

Compound Symbols:

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

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

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

F(ok(z0), ok(z1)) → c4(F(z0, z1))

We considered the (Usable) Rules:

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

And the Tuples:

ACTIVE(f(z0, z1)) → c1(F(active(z0), z1), ACTIVE(z0))
ACTIVE(g(z0)) → c2(G(active(z0)), ACTIVE(z0))
F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(f(z0, z1)) → c7(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(g(z0), z1)) → c(G(z0))

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

POL(ACTIVE(x₁)) = [2]x₁   
POL(F(x₁, x₂)) = x₂   
POL(G(x₁)) = 0   
POL(PROPER(x₁)) = 0   
POL(TOP(x₁)) = [2]x₁ + [2]x₁²   
POL(active(x₁)) = x₁   
POL(c(x₁)) = x₁   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c10(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, x₃)) = x₁ + x₂ + x₃   
POL(c8(x₁, x₂)) = x₁ + x₂   
POL(c9(x₁, x₂)) = x₁ + x₂   
POL(f(x₁, x₂)) = x₁ + [2]x₂   
POL(g(x₁)) = x₁   
POL(mark(x₁)) = 0   
POL(ok(x₁)) = [1] + x₁   
POL(proper(x₁)) = 0   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ACTIVE(f(z0, z1)) → c1(F(active(z0), z1), ACTIVE(z0))
ACTIVE(g(z0)) → c2(G(active(z0)), ACTIVE(z0))
F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(f(z0, z1)) → c7(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(g(z0), z1)) → c(G(z0))

S tuples:

ACTIVE(f(z0, z1)) → c1(F(active(z0), z1), ACTIVE(z0))
ACTIVE(g(z0)) → c2(G(active(z0)), ACTIVE(z0))
F(mark(z0), z1) → c3(F(z0, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(f(z0, z1)) → c7(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))

K tuples:

TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(g(z0), z1)) → c(G(z0))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
F(ok(z0), ok(z1)) → c4(F(z0, z1))

Defined Rule Symbols:

active, f, g, proper, top

Defined Pair Symbols:

ACTIVE, F, G, PROPER, TOP

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, 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.

G(ok(z0)) → c6(G(z0))

We considered the (Usable) Rules:

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

And the Tuples:

ACTIVE(f(z0, z1)) → c1(F(active(z0), z1), ACTIVE(z0))
ACTIVE(g(z0)) → c2(G(active(z0)), ACTIVE(z0))
F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(f(z0, z1)) → c7(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(g(z0), z1)) → c(G(z0))

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

POL(ACTIVE(x₁)) = x₁   
POL(F(x₁, x₂)) = 0   
POL(G(x₁)) = x₁   
POL(PROPER(x₁)) = 0   
POL(TOP(x₁)) = [2]x₁²   
POL(active(x₁)) = x₁   
POL(c(x₁)) = x₁   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c10(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, x₃)) = x₁ + x₂ + x₃   
POL(c8(x₁, x₂)) = x₁ + x₂   
POL(c9(x₁, x₂)) = x₁ + x₂   
POL(f(x₁, x₂)) = x₁   
POL(g(x₁)) = [2]x₁   
POL(mark(x₁)) = x₁   
POL(ok(x₁)) = [1] + x₁   
POL(proper(x₁)) = 0   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ACTIVE(f(z0, z1)) → c1(F(active(z0), z1), ACTIVE(z0))
ACTIVE(g(z0)) → c2(G(active(z0)), ACTIVE(z0))
F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(f(z0, z1)) → c7(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(g(z0), z1)) → c(G(z0))

S tuples:

ACTIVE(f(z0, z1)) → c1(F(active(z0), z1), ACTIVE(z0))
ACTIVE(g(z0)) → c2(G(active(z0)), ACTIVE(z0))
F(mark(z0), z1) → c3(F(z0, z1))
G(mark(z0)) → c5(G(z0))
PROPER(f(z0, z1)) → c7(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))

K tuples:

TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(g(z0), z1)) → c(G(z0))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(ok(z0)) → c6(G(z0))

Defined Rule Symbols:

active, f, g, proper, top

Defined Pair Symbols:

ACTIVE, F, G, PROPER, TOP

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, 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(g(z0)) → c2(G(active(z0)), ACTIVE(z0))

We considered the (Usable) Rules:

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

And the Tuples:

ACTIVE(f(z0, z1)) → c1(F(active(z0), z1), ACTIVE(z0))
ACTIVE(g(z0)) → c2(G(active(z0)), ACTIVE(z0))
F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(f(z0, z1)) → c7(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(g(z0), z1)) → c(G(z0))

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

POL(ACTIVE(x₁)) = [2] + x₁   
POL(F(x₁, x₂)) = 0   
POL(G(x₁)) = 0   
POL(PROPER(x₁)) = 0   
POL(TOP(x₁)) = [2]x₁²   
POL(active(x₁)) = x₁   
POL(c(x₁)) = x₁   
POL(c1(x₁, x₂)) = x₁ + x₂   
POL(c10(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, x₃)) = x₁ + x₂ + x₃   
POL(c8(x₁, x₂)) = x₁ + x₂   
POL(c9(x₁, x₂)) = x₁ + x₂   
POL(f(x₁, x₂)) = [2]x₁   
POL(g(x₁)) = [1] + [2]x₁   
POL(mark(x₁)) = x₁   
POL(ok(x₁)) = [1] + x₁   
POL(proper(x₁)) = x₁   

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ACTIVE(f(z0, z1)) → c1(F(active(z0), z1), ACTIVE(z0))
ACTIVE(g(z0)) → c2(G(active(z0)), ACTIVE(z0))
F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(f(z0, z1)) → c7(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(g(z0), z1)) → c(G(z0))

S tuples:

ACTIVE(f(z0, z1)) → c1(F(active(z0), z1), ACTIVE(z0))
F(mark(z0), z1) → c3(F(z0, z1))
G(mark(z0)) → c5(G(z0))
PROPER(f(z0, z1)) → c7(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))

K tuples:

TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(g(z0), z1)) → c(G(z0))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(ok(z0)) → c6(G(z0))
ACTIVE(g(z0)) → c2(G(active(z0)), ACTIVE(z0))

Defined Rule Symbols:

active, f, g, proper, top

Defined Pair Symbols:

ACTIVE, F, G, PROPER, TOP

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, 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.

ACTIVE(f(z0, z1)) → c1(F(active(z0), z1), ACTIVE(z0))

We considered the (Usable) Rules:

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

And the Tuples:

ACTIVE(f(z0, z1)) → c1(F(active(z0), z1), ACTIVE(z0))
ACTIVE(g(z0)) → c2(G(active(z0)), ACTIVE(z0))
F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(f(z0, z1)) → c7(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(g(z0), z1)) → c(G(z0))

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

POL(ACTIVE(x₁)) = [2]x₁   
POL(F(x₁, x₂)) = 0   
POL(G(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, 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, x₃)) = x₁ + x₂ + x₃   
POL(c8(x₁, x₂)) = x₁ + x₂   
POL(c9(x₁, x₂)) = x₁ + x₂   
POL(f(x₁, x₂)) = [1] + [2]x₁   
POL(g(x₁)) = x₁   
POL(mark(x₁)) = x₁   
POL(ok(x₁)) = [2] + x₁   
POL(proper(x₁)) = x₁   

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ACTIVE(f(z0, z1)) → c1(F(active(z0), z1), ACTIVE(z0))
ACTIVE(g(z0)) → c2(G(active(z0)), ACTIVE(z0))
F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(f(z0, z1)) → c7(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(g(z0), z1)) → c(G(z0))

S tuples:

F(mark(z0), z1) → c3(F(z0, z1))
G(mark(z0)) → c5(G(z0))
PROPER(f(z0, z1)) → c7(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))

K tuples:

TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(g(z0), z1)) → c(G(z0))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(ok(z0)) → c6(G(z0))
ACTIVE(g(z0)) → c2(G(active(z0)), ACTIVE(z0))
ACTIVE(f(z0, z1)) → c1(F(active(z0), z1), ACTIVE(z0))

Defined Rule Symbols:

active, f, g, proper, top

Defined Pair Symbols:

ACTIVE, F, G, PROPER, TOP

Compound Symbols:

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

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

Use narrowing to replace ACTIVE(f(z0, z1)) → c1(F(active(z0), z1), ACTIVE(z0)) by

ACTIVE(f(f(g(z0), z1), x1)) → c1(F(mark(f(z0, f(g(z0), z1))), x1), ACTIVE(f(g(z0), z1)))
ACTIVE(f(f(z0, z1), x1)) → c1(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(z0), x1)) → c1(F(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(f(x0, x1)) → c1

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ACTIVE(g(z0)) → c2(G(active(z0)), ACTIVE(z0))
F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(f(z0, z1)) → c7(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(g(z0), z1)) → c(G(z0))
ACTIVE(f(f(g(z0), z1), x1)) → c1(F(mark(f(z0, f(g(z0), z1))), x1), ACTIVE(f(g(z0), z1)))
ACTIVE(f(f(z0, z1), x1)) → c1(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(z0), x1)) → c1(F(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(f(x0, x1)) → c1

S tuples:

F(mark(z0), z1) → c3(F(z0, z1))
G(mark(z0)) → c5(G(z0))
PROPER(f(z0, z1)) → c7(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))

K tuples:

TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(g(z0), z1)) → c(G(z0))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(ok(z0)) → c6(G(z0))
ACTIVE(g(z0)) → c2(G(active(z0)), ACTIVE(z0))
ACTIVE(f(z0, z1)) → c1(F(active(z0), z1), ACTIVE(z0))

Defined Rule Symbols:

active, f, g, proper, top

Defined Pair Symbols:

ACTIVE, F, G, PROPER, TOP

Compound Symbols:

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

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

Removed 1 trailing nodes:

ACTIVE(f(x0, x1)) → c1

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ACTIVE(g(z0)) → c2(G(active(z0)), ACTIVE(z0))
F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(f(z0, z1)) → c7(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(g(z0), z1)) → c(G(z0))
ACTIVE(f(f(g(z0), z1), x1)) → c1(F(mark(f(z0, f(g(z0), z1))), x1), ACTIVE(f(g(z0), z1)))
ACTIVE(f(f(z0, z1), x1)) → c1(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(z0), x1)) → c1(F(g(active(z0)), x1), ACTIVE(g(z0)))

S tuples:

F(mark(z0), z1) → c3(F(z0, z1))
G(mark(z0)) → c5(G(z0))
PROPER(f(z0, z1)) → c7(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))

K tuples:

TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(g(z0), z1)) → c(G(z0))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(ok(z0)) → c6(G(z0))
ACTIVE(g(z0)) → c2(G(active(z0)), ACTIVE(z0))

Defined Rule Symbols:

active, f, g, proper, top

Defined Pair Symbols:

ACTIVE, F, G, PROPER, TOP

Compound Symbols:

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

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

Use narrowing to replace ACTIVE(g(z0)) → c2(G(active(z0)), ACTIVE(z0)) by

ACTIVE(g(f(g(z0), z1))) → c2(G(mark(f(z0, f(g(z0), z1)))), ACTIVE(f(g(z0), z1)))
ACTIVE(g(f(z0, z1))) → c2(G(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(g(g(z0))) → c2(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(x0)) → c2

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(f(z0, z1)) → c7(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(g(z0), z1)) → c(G(z0))
ACTIVE(f(f(g(z0), z1), x1)) → c1(F(mark(f(z0, f(g(z0), z1))), x1), ACTIVE(f(g(z0), z1)))
ACTIVE(f(f(z0, z1), x1)) → c1(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(z0), x1)) → c1(F(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(g(f(g(z0), z1))) → c2(G(mark(f(z0, f(g(z0), z1)))), ACTIVE(f(g(z0), z1)))
ACTIVE(g(f(z0, z1))) → c2(G(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(g(g(z0))) → c2(G(g(active(z0))), ACTIVE(g(z0)))
ACTIVE(g(x0)) → c2

S tuples:

F(mark(z0), z1) → c3(F(z0, z1))
G(mark(z0)) → c5(G(z0))
PROPER(f(z0, z1)) → c7(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))

K tuples:

TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(g(z0), z1)) → c(G(z0))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(ok(z0)) → c6(G(z0))
ACTIVE(g(z0)) → c2(G(active(z0)), ACTIVE(z0))

Defined Rule Symbols:

active, f, g, proper, top

Defined Pair Symbols:

F, G, PROPER, TOP, ACTIVE

Compound Symbols:

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

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

Removed 1 trailing nodes:

ACTIVE(g(x0)) → c2

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(f(z0, z1)) → c7(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(g(z0), z1)) → c(G(z0))
ACTIVE(f(f(g(z0), z1), x1)) → c1(F(mark(f(z0, f(g(z0), z1))), x1), ACTIVE(f(g(z0), z1)))
ACTIVE(f(f(z0, z1), x1)) → c1(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(z0), x1)) → c1(F(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(g(f(g(z0), z1))) → c2(G(mark(f(z0, f(g(z0), z1)))), ACTIVE(f(g(z0), z1)))
ACTIVE(g(f(z0, z1))) → c2(G(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(g(g(z0))) → c2(G(g(active(z0))), ACTIVE(g(z0)))

S tuples:

F(mark(z0), z1) → c3(F(z0, z1))
G(mark(z0)) → c5(G(z0))
PROPER(f(z0, z1)) → c7(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))

K tuples:

TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(g(z0), z1)) → c(G(z0))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(ok(z0)) → c6(G(z0))

Defined Rule Symbols:

active, f, g, proper, top

Defined Pair Symbols:

F, G, PROPER, TOP, ACTIVE

Compound Symbols:

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

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

Use narrowing to replace PROPER(f(z0, z1)) → c7(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(f(x0, f(z0, z1))) → c7(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(x0, g(z0))) → c7(F(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(f(f(z0, z1), x1)) → c7(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(g(z0), x1)) → c7(F(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(f(x0, x1)) → c7

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(g(z0), z1)) → c(G(z0))
ACTIVE(f(f(g(z0), z1), x1)) → c1(F(mark(f(z0, f(g(z0), z1))), x1), ACTIVE(f(g(z0), z1)))
ACTIVE(f(f(z0, z1), x1)) → c1(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(z0), x1)) → c1(F(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(g(f(g(z0), z1))) → c2(G(mark(f(z0, f(g(z0), z1)))), ACTIVE(f(g(z0), z1)))
ACTIVE(g(f(z0, z1))) → c2(G(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(g(g(z0))) → c2(G(g(active(z0))), ACTIVE(g(z0)))
PROPER(f(x0, f(z0, z1))) → c7(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(x0, g(z0))) → c7(F(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(f(f(z0, z1), x1)) → c7(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(g(z0), x1)) → c7(F(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(f(x0, x1)) → c7

S tuples:

F(mark(z0), z1) → c3(F(z0, z1))
G(mark(z0)) → c5(G(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(f(x0, f(z0, z1))) → c7(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(x0, g(z0))) → c7(F(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(f(f(z0, z1), x1)) → c7(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(g(z0), x1)) → c7(F(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(f(x0, x1)) → c7

K tuples:

TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(g(z0), z1)) → c(G(z0))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(ok(z0)) → c6(G(z0))

Defined Rule Symbols:

active, f, g, proper, top

Defined Pair Symbols:

F, G, PROPER, TOP, ACTIVE

Compound Symbols:

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

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

Removed 1 trailing nodes:

PROPER(f(x0, x1)) → c7

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(g(z0), z1)) → c(G(z0))
ACTIVE(f(f(g(z0), z1), x1)) → c1(F(mark(f(z0, f(g(z0), z1))), x1), ACTIVE(f(g(z0), z1)))
ACTIVE(f(f(z0, z1), x1)) → c1(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(z0), x1)) → c1(F(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(g(f(g(z0), z1))) → c2(G(mark(f(z0, f(g(z0), z1)))), ACTIVE(f(g(z0), z1)))
ACTIVE(g(f(z0, z1))) → c2(G(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(g(g(z0))) → c2(G(g(active(z0))), ACTIVE(g(z0)))
PROPER(f(x0, f(z0, z1))) → c7(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(x0, g(z0))) → c7(F(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(f(f(z0, z1), x1)) → c7(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(g(z0), x1)) → c7(F(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))

S tuples:

F(mark(z0), z1) → c3(F(z0, z1))
G(mark(z0)) → c5(G(z0))
PROPER(g(z0)) → c8(G(proper(z0)), PROPER(z0))
PROPER(f(x0, f(z0, z1))) → c7(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(x0, g(z0))) → c7(F(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(f(f(z0, z1), x1)) → c7(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(g(z0), x1)) → c7(F(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))

K tuples:

TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(g(z0), z1)) → c(G(z0))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(ok(z0)) → c6(G(z0))

Defined Rule Symbols:

active, f, g, proper, top

Defined Pair Symbols:

F, G, PROPER, TOP, ACTIVE

Compound Symbols:

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

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

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

PROPER(g(f(z0, z1))) → c8(G(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(x0)) → c8

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(g(z0), z1)) → c(G(z0))
ACTIVE(f(f(g(z0), z1), x1)) → c1(F(mark(f(z0, f(g(z0), z1))), x1), ACTIVE(f(g(z0), z1)))
ACTIVE(f(f(z0, z1), x1)) → c1(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(z0), x1)) → c1(F(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(g(f(g(z0), z1))) → c2(G(mark(f(z0, f(g(z0), z1)))), ACTIVE(f(g(z0), z1)))
ACTIVE(g(f(z0, z1))) → c2(G(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(g(g(z0))) → c2(G(g(active(z0))), ACTIVE(g(z0)))
PROPER(f(x0, f(z0, z1))) → c7(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(x0, g(z0))) → c7(F(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(f(f(z0, z1), x1)) → c7(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(g(z0), x1)) → c7(F(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(g(f(z0, z1))) → c8(G(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(x0)) → c8

S tuples:

F(mark(z0), z1) → c3(F(z0, z1))
G(mark(z0)) → c5(G(z0))
PROPER(f(x0, f(z0, z1))) → c7(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(x0, g(z0))) → c7(F(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(f(f(z0, z1), x1)) → c7(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(g(z0), x1)) → c7(F(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(g(f(z0, z1))) → c8(G(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))
PROPER(g(x0)) → c8

K tuples:

TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(g(z0), z1)) → c(G(z0))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(ok(z0)) → c6(G(z0))

Defined Rule Symbols:

active, f, g, proper, top

Defined Pair Symbols:

F, G, TOP, ACTIVE, PROPER

Compound Symbols:

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

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

Removed 1 trailing nodes:

PROPER(g(x0)) → c8

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(g(z0), z1)) → c(G(z0))
ACTIVE(f(f(g(z0), z1), x1)) → c1(F(mark(f(z0, f(g(z0), z1))), x1), ACTIVE(f(g(z0), z1)))
ACTIVE(f(f(z0, z1), x1)) → c1(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(z0), x1)) → c1(F(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(g(f(g(z0), z1))) → c2(G(mark(f(z0, f(g(z0), z1)))), ACTIVE(f(g(z0), z1)))
ACTIVE(g(f(z0, z1))) → c2(G(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(g(g(z0))) → c2(G(g(active(z0))), ACTIVE(g(z0)))
PROPER(f(x0, f(z0, z1))) → c7(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(x0, g(z0))) → c7(F(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(f(f(z0, z1), x1)) → c7(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(g(z0), x1)) → c7(F(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(g(f(z0, z1))) → c8(G(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))

S tuples:

F(mark(z0), z1) → c3(F(z0, z1))
G(mark(z0)) → c5(G(z0))
PROPER(f(x0, f(z0, z1))) → c7(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(x0, g(z0))) → c7(F(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(f(f(z0, z1), x1)) → c7(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(g(z0), x1)) → c7(F(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(g(f(z0, z1))) → c8(G(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))

K tuples:

TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(g(z0), z1)) → c(G(z0))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(ok(z0)) → c6(G(z0))

Defined Rule Symbols:

active, f, g, proper, top

Defined Pair Symbols:

F, G, TOP, ACTIVE, PROPER

Compound Symbols:

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

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

Use narrowing to replace TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(f(z0, z1))) → c9(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(g(z0))) → c9(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(x0)) → c9

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(g(z0), z1)) → c(G(z0))
ACTIVE(f(f(g(z0), z1), x1)) → c1(F(mark(f(z0, f(g(z0), z1))), x1), ACTIVE(f(g(z0), z1)))
ACTIVE(f(f(z0, z1), x1)) → c1(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(z0), x1)) → c1(F(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(g(f(g(z0), z1))) → c2(G(mark(f(z0, f(g(z0), z1)))), ACTIVE(f(g(z0), z1)))
ACTIVE(g(f(z0, z1))) → c2(G(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(g(g(z0))) → c2(G(g(active(z0))), ACTIVE(g(z0)))
PROPER(f(x0, f(z0, z1))) → c7(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(x0, g(z0))) → c7(F(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(f(f(z0, z1), x1)) → c7(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(g(z0), x1)) → c7(F(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(g(f(z0, z1))) → c8(G(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))
TOP(mark(f(z0, z1))) → c9(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(g(z0))) → c9(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(mark(x0)) → c9

S tuples:

F(mark(z0), z1) → c3(F(z0, z1))
G(mark(z0)) → c5(G(z0))
PROPER(f(x0, f(z0, z1))) → c7(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(x0, g(z0))) → c7(F(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(f(f(z0, z1), x1)) → c7(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(g(z0), x1)) → c7(F(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(g(f(z0, z1))) → c8(G(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))

K tuples:

TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(g(z0), z1)) → c(G(z0))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(ok(z0)) → c6(G(z0))

Defined Rule Symbols:

active, f, g, proper, top

Defined Pair Symbols:

F, G, TOP, ACTIVE, PROPER

Compound Symbols:

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

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

Removed 1 trailing nodes:

TOP(mark(x0)) → c9

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(g(z0), z1)) → c(G(z0))
ACTIVE(f(f(g(z0), z1), x1)) → c1(F(mark(f(z0, f(g(z0), z1))), x1), ACTIVE(f(g(z0), z1)))
ACTIVE(f(f(z0, z1), x1)) → c1(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(z0), x1)) → c1(F(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(g(f(g(z0), z1))) → c2(G(mark(f(z0, f(g(z0), z1)))), ACTIVE(f(g(z0), z1)))
ACTIVE(g(f(z0, z1))) → c2(G(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(g(g(z0))) → c2(G(g(active(z0))), ACTIVE(g(z0)))
PROPER(f(x0, f(z0, z1))) → c7(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(x0, g(z0))) → c7(F(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(f(f(z0, z1), x1)) → c7(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(g(z0), x1)) → c7(F(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(g(f(z0, z1))) → c8(G(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))
TOP(mark(f(z0, z1))) → c9(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(g(z0))) → c9(TOP(g(proper(z0))), PROPER(g(z0)))

S tuples:

F(mark(z0), z1) → c3(F(z0, z1))
G(mark(z0)) → c5(G(z0))
PROPER(f(x0, f(z0, z1))) → c7(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(x0, g(z0))) → c7(F(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(f(f(z0, z1), x1)) → c7(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(g(z0), x1)) → c7(F(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(g(f(z0, z1))) → c8(G(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))

K tuples:

TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(g(z0), z1)) → c(G(z0))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(ok(z0)) → c6(G(z0))

Defined Rule Symbols:

active, f, g, proper, top

Defined Pair Symbols:

F, G, TOP, ACTIVE, PROPER

Compound Symbols:

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

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

Use narrowing to replace TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(f(g(z0), z1))) → c10(TOP(mark(f(z0, f(g(z0), z1)))), ACTIVE(f(g(z0), z1)))
TOP(ok(f(z0, z1))) → c10(TOP(f(active(z0), z1)), ACTIVE(f(z0, z1)))
TOP(ok(g(z0))) → c10(TOP(g(active(z0))), ACTIVE(g(z0)))
TOP(ok(x0)) → c10

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
ACTIVE(f(g(z0), z1)) → c(G(z0))
ACTIVE(f(f(g(z0), z1), x1)) → c1(F(mark(f(z0, f(g(z0), z1))), x1), ACTIVE(f(g(z0), z1)))
ACTIVE(f(f(z0, z1), x1)) → c1(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(z0), x1)) → c1(F(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(g(f(g(z0), z1))) → c2(G(mark(f(z0, f(g(z0), z1)))), ACTIVE(f(g(z0), z1)))
ACTIVE(g(f(z0, z1))) → c2(G(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(g(g(z0))) → c2(G(g(active(z0))), ACTIVE(g(z0)))
PROPER(f(x0, f(z0, z1))) → c7(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(x0, g(z0))) → c7(F(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(f(f(z0, z1), x1)) → c7(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(g(z0), x1)) → c7(F(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(g(f(z0, z1))) → c8(G(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))
TOP(mark(f(z0, z1))) → c9(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(g(z0))) → c9(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(ok(f(g(z0), z1))) → c10(TOP(mark(f(z0, f(g(z0), z1)))), ACTIVE(f(g(z0), z1)))
TOP(ok(f(z0, z1))) → c10(TOP(f(active(z0), z1)), ACTIVE(f(z0, z1)))
TOP(ok(g(z0))) → c10(TOP(g(active(z0))), ACTIVE(g(z0)))
TOP(ok(x0)) → c10

S tuples:

F(mark(z0), z1) → c3(F(z0, z1))
G(mark(z0)) → c5(G(z0))
PROPER(f(x0, f(z0, z1))) → c7(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(x0, g(z0))) → c7(F(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(f(f(z0, z1), x1)) → c7(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(g(z0), x1)) → c7(F(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(g(f(z0, z1))) → c8(G(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))

K tuples:

TOP(ok(z0)) → c10(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(g(z0), z1)) → c(G(z0))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(ok(z0)) → c6(G(z0))

Defined Rule Symbols:

active, f, g, proper, top

Defined Pair Symbols:

F, G, ACTIVE, PROPER, TOP

Compound Symbols:

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

(39) CdtUnreachableProof (EQUIVALENT transformation)

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

ACTIVE(f(g(z0), z1)) → c(G(z0))
ACTIVE(f(f(g(z0), z1), x1)) → c1(F(mark(f(z0, f(g(z0), z1))), x1), ACTIVE(f(g(z0), z1)))
ACTIVE(f(f(z0, z1), x1)) → c1(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(g(z0), x1)) → c1(F(g(active(z0)), x1), ACTIVE(g(z0)))
ACTIVE(g(f(g(z0), z1))) → c2(G(mark(f(z0, f(g(z0), z1)))), ACTIVE(f(g(z0), z1)))
ACTIVE(g(f(z0, z1))) → c2(G(f(active(z0), z1)), ACTIVE(f(z0, z1)))
ACTIVE(g(g(z0))) → c2(G(g(active(z0))), ACTIVE(g(z0)))
PROPER(f(x0, f(z0, z1))) → c7(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(x0, g(z0))) → c7(F(proper(x0), g(proper(z0))), PROPER(x0), PROPER(g(z0)))
PROPER(f(f(z0, z1), x1)) → c7(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(g(z0), x1)) → c7(F(g(proper(z0)), proper(x1)), PROPER(g(z0)), PROPER(x1))
PROPER(g(f(z0, z1))) → c8(G(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
PROPER(g(g(z0))) → c8(G(g(proper(z0))), PROPER(g(z0)))
TOP(mark(f(z0, z1))) → c9(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(g(z0))) → c9(TOP(g(proper(z0))), PROPER(g(z0)))
TOP(ok(f(g(z0), z1))) → c10(TOP(mark(f(z0, f(g(z0), z1)))), ACTIVE(f(g(z0), z1)))
TOP(ok(f(z0, z1))) → c10(TOP(f(active(z0), z1)), ACTIVE(f(z0, z1)))
TOP(ok(g(z0))) → c10(TOP(g(active(z0))), ACTIVE(g(z0)))

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))
TOP(ok(x0)) → c10

S tuples:

F(mark(z0), z1) → c3(F(z0, z1))
G(mark(z0)) → c5(G(z0))

K tuples:

F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(ok(z0)) → c6(G(z0))

Defined Rule Symbols:

active, f, g, proper, top

Defined Pair Symbols:

F, G, TOP

Compound Symbols:

c3, c4, c5, c6, c10

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

Removed 1 trailing nodes:

TOP(ok(x0)) → c10

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))

S tuples:

F(mark(z0), z1) → c3(F(z0, z1))
G(mark(z0)) → c5(G(z0))

K tuples:

F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(ok(z0)) → c6(G(z0))

Defined Rule Symbols:

active, f, g, proper, top

Defined Pair Symbols:

F, G

Compound Symbols:

c3, c4, c5, c6

(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.

G(mark(z0)) → c5(G(z0))

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

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))

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

POL(F(x₁, x₂)) = [2]x₂   
POL(G(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(ok(x₁)) = x₁   

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))

S tuples:

F(mark(z0), z1) → c3(F(z0, z1))

K tuples:

F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(ok(z0)) → c6(G(z0))
G(mark(z0)) → c5(G(z0))

Defined Rule Symbols:

active, f, g, proper, top

Defined Pair Symbols:

F, G

Compound Symbols:

c3, c4, c5, c6

(45) 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.

F(mark(z0), z1) → c3(F(z0, z1))

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

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))

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

POL(F(x₁, x₂)) = x₁   
POL(G(x₁)) = 0   
POL(c3(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(ok(x₁)) = x₁   

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(mark(z0)) → c5(G(z0))
G(ok(z0)) → c6(G(z0))

S tuples:none
K tuples:

F(ok(z0), ok(z1)) → c4(F(z0, z1))
G(ok(z0)) → c6(G(z0))
G(mark(z0)) → c5(G(z0))
F(mark(z0), z1) → c3(F(z0, z1))

Defined Rule Symbols:

active, f, g, proper, top

Defined Pair Symbols:

F, G

Compound Symbols:

c3, c4, c5, c6

(47) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(48) BOUNDS(O(1), O(1))