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

(0) Obligation:

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

active(f(x)) → mark(f(f(x)))
chk(no(f(x))) → f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))
mat(f(x), f(y)) → f(mat(x, y))
chk(no(c)) → active(c)
mat(f(x), c) → no(c)
f(active(x)) → active(f(x))
f(no(x)) → no(f(x))
f(mark(x)) → mark(f(x))
tp(mark(x)) → tp(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), x)))

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(f(f(z0)))
chk(no(f(z0))) → f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)))
chk(no(c)) → active(c)
mat(f(z0), f(y)) → f(mat(z0, y))
mat(f(z0), c) → no(c)
f(active(z0)) → active(f(z0))
f(no(z0)) → no(f(z0))
f(mark(z0)) → mark(f(z0))
tp(mark(z0)) → tp(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)))

Tuples:

ACTIVE(f(z0)) → c1(F(f(z0)), F(z0))
CHK(no(f(z0))) → c2(F(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0), F(f(f(f(f(f(f(f(f(f(X)))))))))), F(f(f(f(f(f(f(f(f(X))))))))), F(f(f(f(f(f(f(f(X)))))))), F(f(f(f(f(f(f(X))))))), F(f(f(f(f(f(X)))))), F(f(f(f(f(X))))), F(f(f(f(X)))), F(f(f(X))), F(f(X)), F(X))
CHK(no(c)) → c3(ACTIVE(c))
MAT(f(z0), f(y)) → c4(F(mat(z0, y)), MAT(z0, y))
MAT(f(z0), c) → c5
F(active(z0)) → c6(ACTIVE(f(z0)), F(z0))
F(no(z0)) → c7(F(z0))
F(mark(z0)) → c8(F(z0))
TP(mark(z0)) → c9(TP(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0), F(f(f(f(f(f(f(f(f(f(X)))))))))), F(f(f(f(f(f(f(f(f(X))))))))), F(f(f(f(f(f(f(f(X)))))))), F(f(f(f(f(f(f(X))))))), F(f(f(f(f(f(X)))))), F(f(f(f(f(X))))), F(f(f(f(X)))), F(f(f(X))), F(f(X)), F(X))

S tuples:

ACTIVE(f(z0)) → c1(F(f(z0)), F(z0))
CHK(no(f(z0))) → c2(F(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0), F(f(f(f(f(f(f(f(f(f(X)))))))))), F(f(f(f(f(f(f(f(f(X))))))))), F(f(f(f(f(f(f(f(X)))))))), F(f(f(f(f(f(f(X))))))), F(f(f(f(f(f(X)))))), F(f(f(f(f(X))))), F(f(f(f(X)))), F(f(f(X))), F(f(X)), F(X))
CHK(no(c)) → c3(ACTIVE(c))
MAT(f(z0), f(y)) → c4(F(mat(z0, y)), MAT(z0, y))
MAT(f(z0), c) → c5
F(active(z0)) → c6(ACTIVE(f(z0)), F(z0))
F(no(z0)) → c7(F(z0))
F(mark(z0)) → c8(F(z0))
TP(mark(z0)) → c9(TP(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0), F(f(f(f(f(f(f(f(f(f(X)))))))))), F(f(f(f(f(f(f(f(f(X))))))))), F(f(f(f(f(f(f(f(X)))))))), F(f(f(f(f(f(f(X))))))), F(f(f(f(f(f(X)))))), F(f(f(f(f(X))))), F(f(f(f(X)))), F(f(f(X))), F(f(X)), F(X))

K tuples:none
Defined Rule Symbols:

active, chk, mat, f, tp

Defined Pair Symbols:

ACTIVE, CHK, MAT, F, TP

Compound Symbols:

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

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

Removed 3 trailing nodes:

CHK(no(c)) → c3(ACTIVE(c))
MAT(f(z0), f(y)) → c4(F(mat(z0, y)), MAT(z0, y))
MAT(f(z0), c) → c5

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(f(f(z0)))
chk(no(f(z0))) → f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)))
chk(no(c)) → active(c)
mat(f(z0), f(y)) → f(mat(z0, y))
mat(f(z0), c) → no(c)
f(active(z0)) → active(f(z0))
f(no(z0)) → no(f(z0))
f(mark(z0)) → mark(f(z0))
tp(mark(z0)) → tp(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)))

Tuples:

ACTIVE(f(z0)) → c1(F(f(z0)), F(z0))
CHK(no(f(z0))) → c2(F(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0), F(f(f(f(f(f(f(f(f(f(X)))))))))), F(f(f(f(f(f(f(f(f(X))))))))), F(f(f(f(f(f(f(f(X)))))))), F(f(f(f(f(f(f(X))))))), F(f(f(f(f(f(X)))))), F(f(f(f(f(X))))), F(f(f(f(X)))), F(f(f(X))), F(f(X)), F(X))
F(active(z0)) → c6(ACTIVE(f(z0)), F(z0))
F(no(z0)) → c7(F(z0))
F(mark(z0)) → c8(F(z0))
TP(mark(z0)) → c9(TP(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0), F(f(f(f(f(f(f(f(f(f(X)))))))))), F(f(f(f(f(f(f(f(f(X))))))))), F(f(f(f(f(f(f(f(X)))))))), F(f(f(f(f(f(f(X))))))), F(f(f(f(f(f(X)))))), F(f(f(f(f(X))))), F(f(f(f(X)))), F(f(f(X))), F(f(X)), F(X))

S tuples:

ACTIVE(f(z0)) → c1(F(f(z0)), F(z0))
CHK(no(f(z0))) → c2(F(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0), F(f(f(f(f(f(f(f(f(f(X)))))))))), F(f(f(f(f(f(f(f(f(X))))))))), F(f(f(f(f(f(f(f(X)))))))), F(f(f(f(f(f(f(X))))))), F(f(f(f(f(f(X)))))), F(f(f(f(f(X))))), F(f(f(f(X)))), F(f(f(X))), F(f(X)), F(X))
F(active(z0)) → c6(ACTIVE(f(z0)), F(z0))
F(no(z0)) → c7(F(z0))
F(mark(z0)) → c8(F(z0))
TP(mark(z0)) → c9(TP(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0), F(f(f(f(f(f(f(f(f(f(X)))))))))), F(f(f(f(f(f(f(f(f(X))))))))), F(f(f(f(f(f(f(f(X)))))))), F(f(f(f(f(f(f(X))))))), F(f(f(f(f(f(X)))))), F(f(f(f(f(X))))), F(f(f(f(X)))), F(f(f(X))), F(f(X)), F(X))

K tuples:none
Defined Rule Symbols:

active, chk, mat, f, tp

Defined Pair Symbols:

ACTIVE, CHK, F, TP

Compound Symbols:

c1, c2, c6, c7, c8, c9

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

Removed 23 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(f(f(z0)))
chk(no(f(z0))) → f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)))
chk(no(c)) → active(c)
mat(f(z0), f(y)) → f(mat(z0, y))
mat(f(z0), c) → no(c)
f(active(z0)) → active(f(z0))
f(no(z0)) → no(f(z0))
f(mark(z0)) → mark(f(z0))
tp(mark(z0)) → tp(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)))

Tuples:

F(active(z0)) → c6(ACTIVE(f(z0)), F(z0))
F(no(z0)) → c7(F(z0))
F(mark(z0)) → c8(F(z0))
ACTIVE(f(z0)) → c1(F(z0))
CHK(no(f(z0))) → c2(F(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)))
TP(mark(z0)) → c9(TP(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)))

S tuples:

F(active(z0)) → c6(ACTIVE(f(z0)), F(z0))
F(no(z0)) → c7(F(z0))
F(mark(z0)) → c8(F(z0))
ACTIVE(f(z0)) → c1(F(z0))
CHK(no(f(z0))) → c2(F(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)))
TP(mark(z0)) → c9(TP(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)))

K tuples:none
Defined Rule Symbols:

active, chk, mat, f, tp

Defined Pair Symbols:

F, ACTIVE, CHK, TP

Compound Symbols:

c6, c7, c8, c1, c2, c9

(7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

tp(mark(z0)) → tp(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(active(z0)) → active(f(z0))
f(no(z0)) → no(f(z0))
f(mark(z0)) → mark(f(z0))
active(f(z0)) → mark(f(f(z0)))
chk(no(f(z0))) → f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)))
chk(no(c)) → active(c)
mat(f(z0), f(y)) → f(mat(z0, y))
mat(f(z0), c) → no(c)

Tuples:

F(active(z0)) → c6(ACTIVE(f(z0)), F(z0))
F(no(z0)) → c7(F(z0))
F(mark(z0)) → c8(F(z0))
ACTIVE(f(z0)) → c1(F(z0))
CHK(no(f(z0))) → c2(F(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)))
TP(mark(z0)) → c9(TP(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)))

S tuples:

F(active(z0)) → c6(ACTIVE(f(z0)), F(z0))
F(no(z0)) → c7(F(z0))
F(mark(z0)) → c8(F(z0))
ACTIVE(f(z0)) → c1(F(z0))
CHK(no(f(z0))) → c2(F(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)))
TP(mark(z0)) → c9(TP(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)))

K tuples:none
Defined Rule Symbols:

f, active, chk, mat

Defined Pair Symbols:

F, ACTIVE, CHK, TP

Compound Symbols:

c6, c7, c8, c1, c2, c9

(9) CdtRuleRemovalProof (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(no(z0)) → c7(F(z0))

We considered the (Usable) Rules:

f(no(z0)) → no(f(z0))
active(f(z0)) → mark(f(f(z0)))
f(active(z0)) → active(f(z0))
f(mark(z0)) → mark(f(z0))
chk(no(c)) → active(c)
chk(no(f(z0))) → f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)))

And the Tuples:

F(active(z0)) → c6(ACTIVE(f(z0)), F(z0))
F(no(z0)) → c7(F(z0))
F(mark(z0)) → c8(F(z0))
ACTIVE(f(z0)) → c1(F(z0))
CHK(no(f(z0))) → c2(F(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)))
TP(mark(z0)) → c9(TP(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)))

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

POL(ACTIVE(x₁)) = [4]x₁   
POL(CHK(x₁)) = 0   
POL(F(x₁)) = [4]x₁   
POL(TP(x₁)) = 0   
POL(X) = 0   
POL(active(x₁)) = [5]x₁   
POL(c) = 0   
POL(c1(x₁)) = x₁   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c6(x₁, x₂)) = x₁ + x₂   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁, x₂)) = x₁ + x₂   
POL(chk(x₁)) = 0   
POL(f(x₁)) = [4]x₁   
POL(mark(x₁)) = x₁   
POL(mat(x₁, x₂)) = 0   
POL(no(x₁)) = [4] + x₁   
POL(y) = 0   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(active(z0)) → active(f(z0))
f(no(z0)) → no(f(z0))
f(mark(z0)) → mark(f(z0))
active(f(z0)) → mark(f(f(z0)))
chk(no(f(z0))) → f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)))
chk(no(c)) → active(c)
mat(f(z0), f(y)) → f(mat(z0, y))
mat(f(z0), c) → no(c)

Tuples:

F(active(z0)) → c6(ACTIVE(f(z0)), F(z0))
F(no(z0)) → c7(F(z0))
F(mark(z0)) → c8(F(z0))
ACTIVE(f(z0)) → c1(F(z0))
CHK(no(f(z0))) → c2(F(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)))
TP(mark(z0)) → c9(TP(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)))

S tuples:

F(active(z0)) → c6(ACTIVE(f(z0)), F(z0))
F(mark(z0)) → c8(F(z0))
ACTIVE(f(z0)) → c1(F(z0))
CHK(no(f(z0))) → c2(F(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)))
TP(mark(z0)) → c9(TP(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)))

K tuples:

F(no(z0)) → c7(F(z0))

Defined Rule Symbols:

f, active, chk, mat

Defined Pair Symbols:

F, ACTIVE, CHK, TP

Compound Symbols:

c6, c7, c8, c1, c2, c9

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

Use narrowing to replace CHK(no(f(z0))) → c2(F(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0))) by

CHK(no(f(f(y)))) → c2(F(chk(f(mat(f(f(f(f(f(f(f(f(f(X))))))))), y)))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))
CHK(no(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(active(z0)) → active(f(z0))
f(no(z0)) → no(f(z0))
f(mark(z0)) → mark(f(z0))
active(f(z0)) → mark(f(f(z0)))
chk(no(f(z0))) → f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)))
chk(no(c)) → active(c)
mat(f(z0), f(y)) → f(mat(z0, y))
mat(f(z0), c) → no(c)

Tuples:

F(active(z0)) → c6(ACTIVE(f(z0)), F(z0))
F(no(z0)) → c7(F(z0))
F(mark(z0)) → c8(F(z0))
ACTIVE(f(z0)) → c1(F(z0))
TP(mark(z0)) → c9(TP(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)))
CHK(no(f(f(y)))) → c2(F(chk(f(mat(f(f(f(f(f(f(f(f(f(X))))))))), y)))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))
CHK(no(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))

S tuples:

F(active(z0)) → c6(ACTIVE(f(z0)), F(z0))
F(mark(z0)) → c8(F(z0))
ACTIVE(f(z0)) → c1(F(z0))
TP(mark(z0)) → c9(TP(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)))
CHK(no(f(f(y)))) → c2(F(chk(f(mat(f(f(f(f(f(f(f(f(f(X))))))))), y)))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))
CHK(no(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))

K tuples:

F(no(z0)) → c7(F(z0))

Defined Rule Symbols:

f, active, chk, mat

Defined Pair Symbols:

F, ACTIVE, TP, CHK

Compound Symbols:

c6, c7, c8, c1, c9, c2

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

Removed 1 trailing tuple parts

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(active(z0)) → active(f(z0))
f(no(z0)) → no(f(z0))
f(mark(z0)) → mark(f(z0))
active(f(z0)) → mark(f(f(z0)))
chk(no(f(z0))) → f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)))
chk(no(c)) → active(c)
mat(f(z0), f(y)) → f(mat(z0, y))
mat(f(z0), c) → no(c)

Tuples:

F(active(z0)) → c6(ACTIVE(f(z0)), F(z0))
F(no(z0)) → c7(F(z0))
F(mark(z0)) → c8(F(z0))
ACTIVE(f(z0)) → c1(F(z0))
TP(mark(z0)) → c9(TP(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)))
CHK(no(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))
CHK(no(f(f(y)))) → c2(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))

S tuples:

F(active(z0)) → c6(ACTIVE(f(z0)), F(z0))
F(mark(z0)) → c8(F(z0))
ACTIVE(f(z0)) → c1(F(z0))
TP(mark(z0)) → c9(TP(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)))
CHK(no(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))
CHK(no(f(f(y)))) → c2(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))

K tuples:

F(no(z0)) → c7(F(z0))

Defined Rule Symbols:

f, active, chk, mat

Defined Pair Symbols:

F, ACTIVE, TP, CHK

Compound Symbols:

c6, c7, c8, c1, c9, c2, c2

(15) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

CHK(no(f(f(y)))) → c2(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))

We considered the (Usable) Rules:

mat(f(z0), c) → no(c)
mat(f(z0), f(y)) → f(mat(z0, y))

And the Tuples:

F(active(z0)) → c6(ACTIVE(f(z0)), F(z0))
F(no(z0)) → c7(F(z0))
F(mark(z0)) → c8(F(z0))
ACTIVE(f(z0)) → c1(F(z0))
TP(mark(z0)) → c9(TP(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)))
CHK(no(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))
CHK(no(f(f(y)))) → c2(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))

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

POL(ACTIVE(x₁)) = 0   
POL(CHK(x₁)) = [4]x₁   
POL(F(x₁)) = 0   
POL(TP(x₁)) = 0   
POL(X) = 0   
POL(active(x₁)) = 0   
POL(c) = 0   
POL(c1(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c6(x₁, x₂)) = x₁ + x₂   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁, x₂)) = x₁ + x₂   
POL(chk(x₁)) = 0   
POL(f(x₁)) = [2]x₁   
POL(mark(x₁)) = 0   
POL(mat(x₁, x₂)) = 0   
POL(no(x₁)) = x₁   
POL(y) = [2]   

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(active(z0)) → active(f(z0))
f(no(z0)) → no(f(z0))
f(mark(z0)) → mark(f(z0))
active(f(z0)) → mark(f(f(z0)))
chk(no(f(z0))) → f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)))
chk(no(c)) → active(c)
mat(f(z0), f(y)) → f(mat(z0, y))
mat(f(z0), c) → no(c)

Tuples:

F(active(z0)) → c6(ACTIVE(f(z0)), F(z0))
F(no(z0)) → c7(F(z0))
F(mark(z0)) → c8(F(z0))
ACTIVE(f(z0)) → c1(F(z0))
TP(mark(z0)) → c9(TP(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)))
CHK(no(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))
CHK(no(f(f(y)))) → c2(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))

S tuples:

F(active(z0)) → c6(ACTIVE(f(z0)), F(z0))
F(mark(z0)) → c8(F(z0))
ACTIVE(f(z0)) → c1(F(z0))
TP(mark(z0)) → c9(TP(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)))
CHK(no(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))

K tuples:

F(no(z0)) → c7(F(z0))
CHK(no(f(f(y)))) → c2(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))

Defined Rule Symbols:

f, active, chk, mat

Defined Pair Symbols:

F, ACTIVE, TP, CHK

Compound Symbols:

c6, c7, c8, c1, c9, c2, c2

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

Use narrowing to replace TP(mark(z0)) → c9(TP(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0))) by

TP(mark(f(y))) → c9(TP(chk(f(mat(f(f(f(f(f(f(f(f(f(X))))))))), y)))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))
TP(mark(c)) → c9(TP(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(active(z0)) → active(f(z0))
f(no(z0)) → no(f(z0))
f(mark(z0)) → mark(f(z0))
active(f(z0)) → mark(f(f(z0)))
chk(no(f(z0))) → f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)))
chk(no(c)) → active(c)
mat(f(z0), f(y)) → f(mat(z0, y))
mat(f(z0), c) → no(c)

Tuples:

F(active(z0)) → c6(ACTIVE(f(z0)), F(z0))
F(no(z0)) → c7(F(z0))
F(mark(z0)) → c8(F(z0))
ACTIVE(f(z0)) → c1(F(z0))
CHK(no(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))
CHK(no(f(f(y)))) → c2(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))
TP(mark(f(y))) → c9(TP(chk(f(mat(f(f(f(f(f(f(f(f(f(X))))))))), y)))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))
TP(mark(c)) → c9(TP(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))

S tuples:

F(active(z0)) → c6(ACTIVE(f(z0)), F(z0))
F(mark(z0)) → c8(F(z0))
ACTIVE(f(z0)) → c1(F(z0))
CHK(no(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))
TP(mark(f(y))) → c9(TP(chk(f(mat(f(f(f(f(f(f(f(f(f(X))))))))), y)))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))
TP(mark(c)) → c9(TP(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))

K tuples:

F(no(z0)) → c7(F(z0))
CHK(no(f(f(y)))) → c2(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))

Defined Rule Symbols:

f, active, chk, mat

Defined Pair Symbols:

F, ACTIVE, CHK, TP

Compound Symbols:

c6, c7, c8, c1, c2, c2, c9

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

Removed 1 trailing tuple parts

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(active(z0)) → active(f(z0))
f(no(z0)) → no(f(z0))
f(mark(z0)) → mark(f(z0))
active(f(z0)) → mark(f(f(z0)))
chk(no(f(z0))) → f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)))
chk(no(c)) → active(c)
mat(f(z0), f(y)) → f(mat(z0, y))
mat(f(z0), c) → no(c)

Tuples:

F(active(z0)) → c6(ACTIVE(f(z0)), F(z0))
F(no(z0)) → c7(F(z0))
F(mark(z0)) → c8(F(z0))
ACTIVE(f(z0)) → c1(F(z0))
CHK(no(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))
CHK(no(f(f(y)))) → c2(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))
TP(mark(c)) → c9(TP(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))
TP(mark(f(y))) → c9(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))

S tuples:

F(active(z0)) → c6(ACTIVE(f(z0)), F(z0))
F(mark(z0)) → c8(F(z0))
ACTIVE(f(z0)) → c1(F(z0))
CHK(no(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))
TP(mark(c)) → c9(TP(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))
TP(mark(f(y))) → c9(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))

K tuples:

F(no(z0)) → c7(F(z0))
CHK(no(f(f(y)))) → c2(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))

Defined Rule Symbols:

f, active, chk, mat

Defined Pair Symbols:

F, ACTIVE, CHK, TP

Compound Symbols:

c6, c7, c8, c1, c2, c2, c9, c9

(21) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

chk(no(f(z0))) → f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)))

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(active(z0)) → active(f(z0))
f(no(z0)) → no(f(z0))
f(mark(z0)) → mark(f(z0))
active(f(z0)) → mark(f(f(z0)))
chk(no(c)) → active(c)
mat(f(z0), c) → no(c)
mat(f(z0), f(y)) → f(mat(z0, y))

Tuples:

F(active(z0)) → c6(ACTIVE(f(z0)), F(z0))
F(no(z0)) → c7(F(z0))
F(mark(z0)) → c8(F(z0))
ACTIVE(f(z0)) → c1(F(z0))
CHK(no(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))
CHK(no(f(f(y)))) → c2(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))
TP(mark(c)) → c9(TP(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))
TP(mark(f(y))) → c9(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))

S tuples:

F(active(z0)) → c6(ACTIVE(f(z0)), F(z0))
F(mark(z0)) → c8(F(z0))
ACTIVE(f(z0)) → c1(F(z0))
CHK(no(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))
TP(mark(c)) → c9(TP(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))
TP(mark(f(y))) → c9(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))

K tuples:

F(no(z0)) → c7(F(z0))
CHK(no(f(f(y)))) → c2(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))

Defined Rule Symbols:

f, active, chk, mat

Defined Pair Symbols:

F, ACTIVE, CHK, TP

Compound Symbols:

c6, c7, c8, c1, c2, c2, c9, c9

(23) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

TP(mark(f(y))) → c9(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))

We considered the (Usable) Rules:

chk(no(c)) → active(c)

And the Tuples:

F(active(z0)) → c6(ACTIVE(f(z0)), F(z0))
F(no(z0)) → c7(F(z0))
F(mark(z0)) → c8(F(z0))
ACTIVE(f(z0)) → c1(F(z0))
CHK(no(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))
CHK(no(f(f(y)))) → c2(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))
TP(mark(c)) → c9(TP(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))
TP(mark(f(y))) → c9(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))

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

POL(ACTIVE(x₁)) = 0   
POL(CHK(x₁)) = 0   
POL(F(x₁)) = 0   
POL(TP(x₁)) = [1] + [2]x₁   
POL(X) = [2]   
POL(active(x₁)) = 0   
POL(c) = [3]   
POL(c1(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c6(x₁, x₂)) = x₁ + x₂   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(c9(x₁, x₂)) = x₁ + x₂   
POL(chk(x₁)) = [3]x₁   
POL(f(x₁)) = [4]x₁   
POL(mark(x₁)) = 0   
POL(mat(x₁, x₂)) = [4] + [5]x₁ + [2]x₂   
POL(no(x₁)) = 0   
POL(y) = 0   

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(active(z0)) → active(f(z0))
f(no(z0)) → no(f(z0))
f(mark(z0)) → mark(f(z0))
active(f(z0)) → mark(f(f(z0)))
chk(no(c)) → active(c)
mat(f(z0), c) → no(c)
mat(f(z0), f(y)) → f(mat(z0, y))

Tuples:

F(active(z0)) → c6(ACTIVE(f(z0)), F(z0))
F(no(z0)) → c7(F(z0))
F(mark(z0)) → c8(F(z0))
ACTIVE(f(z0)) → c1(F(z0))
CHK(no(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))
CHK(no(f(f(y)))) → c2(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))
TP(mark(c)) → c9(TP(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))
TP(mark(f(y))) → c9(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))

S tuples:

F(active(z0)) → c6(ACTIVE(f(z0)), F(z0))
F(mark(z0)) → c8(F(z0))
ACTIVE(f(z0)) → c1(F(z0))
CHK(no(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))
TP(mark(c)) → c9(TP(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))

K tuples:

F(no(z0)) → c7(F(z0))
CHK(no(f(f(y)))) → c2(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))
TP(mark(f(y))) → c9(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))

Defined Rule Symbols:

f, active, chk, mat

Defined Pair Symbols:

F, ACTIVE, CHK, TP

Compound Symbols:

c6, c7, c8, c1, c2, c2, c9, c9

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

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

TP(mark(c)) → c9(TP(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))

We considered the (Usable) Rules:

chk(no(c)) → active(c)

And the Tuples:

F(active(z0)) → c6(ACTIVE(f(z0)), F(z0))
F(no(z0)) → c7(F(z0))
F(mark(z0)) → c8(F(z0))
ACTIVE(f(z0)) → c1(F(z0))
CHK(no(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))
CHK(no(f(f(y)))) → c2(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))
TP(mark(c)) → c9(TP(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))
TP(mark(f(y))) → c9(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))

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

POL(ACTIVE(x₁)) = 0   
POL(CHK(x₁)) = 0   
POL(F(x₁)) = 0   
POL(TP(x₁)) = [1] + [4]x₁   
POL(X) = [3]   
POL(active(x₁)) = [3] + [3]x₁   
POL(c) = 0   
POL(c1(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c6(x₁, x₂)) = x₁ + x₂   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(c9(x₁, x₂)) = x₁ + x₂   
POL(chk(x₁)) = [4] + x₁   
POL(f(x₁)) = [2] + [2]x₁   
POL(mark(x₁)) = [5]   
POL(mat(x₁, x₂)) = [5] + [2]x₁ + [4]x₂   
POL(no(x₁)) = 0   
POL(y) = 0   

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(active(z0)) → active(f(z0))
f(no(z0)) → no(f(z0))
f(mark(z0)) → mark(f(z0))
active(f(z0)) → mark(f(f(z0)))
chk(no(c)) → active(c)
mat(f(z0), c) → no(c)
mat(f(z0), f(y)) → f(mat(z0, y))

Tuples:

F(active(z0)) → c6(ACTIVE(f(z0)), F(z0))
F(no(z0)) → c7(F(z0))
F(mark(z0)) → c8(F(z0))
ACTIVE(f(z0)) → c1(F(z0))
CHK(no(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))
CHK(no(f(f(y)))) → c2(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))
TP(mark(c)) → c9(TP(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))
TP(mark(f(y))) → c9(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))

S tuples:

F(active(z0)) → c6(ACTIVE(f(z0)), F(z0))
F(mark(z0)) → c8(F(z0))
ACTIVE(f(z0)) → c1(F(z0))
CHK(no(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))

K tuples:

F(no(z0)) → c7(F(z0))
CHK(no(f(f(y)))) → c2(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))
TP(mark(f(y))) → c9(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))
TP(mark(c)) → c9(TP(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))

Defined Rule Symbols:

f, active, chk, mat

Defined Pair Symbols:

F, ACTIVE, CHK, TP

Compound Symbols:

c6, c7, c8, c1, c2, c2, c9, c9

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

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

CHK(no(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))

We considered the (Usable) Rules:

mat(f(z0), c) → no(c)
mat(f(z0), f(y)) → f(mat(z0, y))

And the Tuples:

F(active(z0)) → c6(ACTIVE(f(z0)), F(z0))
F(no(z0)) → c7(F(z0))
F(mark(z0)) → c8(F(z0))
ACTIVE(f(z0)) → c1(F(z0))
CHK(no(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))
CHK(no(f(f(y)))) → c2(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))
TP(mark(c)) → c9(TP(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))
TP(mark(f(y))) → c9(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))

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

POL(ACTIVE(x₁)) = 0   
POL(CHK(x₁)) = x₁   
POL(F(x₁)) = 0   
POL(TP(x₁)) = [1]   
POL(X) = 0   
POL(active(x₁)) = 0   
POL(c) = 0   
POL(c1(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c6(x₁, x₂)) = x₁ + x₂   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(c9(x₁, x₂)) = x₁ + x₂   
POL(chk(x₁)) = [4]   
POL(f(x₁)) = [1]   
POL(mark(x₁)) = 0   
POL(mat(x₁, x₂)) = x₂   
POL(no(x₁)) = x₁   
POL(y) = 0   

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(active(z0)) → active(f(z0))
f(no(z0)) → no(f(z0))
f(mark(z0)) → mark(f(z0))
active(f(z0)) → mark(f(f(z0)))
chk(no(c)) → active(c)
mat(f(z0), c) → no(c)
mat(f(z0), f(y)) → f(mat(z0, y))

Tuples:

F(active(z0)) → c6(ACTIVE(f(z0)), F(z0))
F(no(z0)) → c7(F(z0))
F(mark(z0)) → c8(F(z0))
ACTIVE(f(z0)) → c1(F(z0))
CHK(no(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))
CHK(no(f(f(y)))) → c2(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))
TP(mark(c)) → c9(TP(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))
TP(mark(f(y))) → c9(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))

S tuples:

F(active(z0)) → c6(ACTIVE(f(z0)), F(z0))
F(mark(z0)) → c8(F(z0))
ACTIVE(f(z0)) → c1(F(z0))

K tuples:

F(no(z0)) → c7(F(z0))
CHK(no(f(f(y)))) → c2(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))
TP(mark(f(y))) → c9(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))
TP(mark(c)) → c9(TP(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))
CHK(no(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))

Defined Rule Symbols:

f, active, chk, mat

Defined Pair Symbols:

F, ACTIVE, CHK, TP

Compound Symbols:

c6, c7, c8, c1, c2, c2, c9, c9

(29) CdtRuleRemovalProof (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(active(z0)) → c6(ACTIVE(f(z0)), F(z0))
F(mark(z0)) → c8(F(z0))
ACTIVE(f(z0)) → c1(F(z0))

We considered the (Usable) Rules:

f(no(z0)) → no(f(z0))
active(f(z0)) → mark(f(f(z0)))
mat(f(z0), c) → no(c)
f(active(z0)) → active(f(z0))
f(mark(z0)) → mark(f(z0))
chk(no(c)) → active(c)
mat(f(z0), f(y)) → f(mat(z0, y))

And the Tuples:

F(active(z0)) → c6(ACTIVE(f(z0)), F(z0))
F(no(z0)) → c7(F(z0))
F(mark(z0)) → c8(F(z0))
ACTIVE(f(z0)) → c1(F(z0))
CHK(no(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))
CHK(no(f(f(y)))) → c2(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))
TP(mark(c)) → c9(TP(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))
TP(mark(f(y))) → c9(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))

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

POL(ACTIVE(x₁)) = [2]x₁   
POL(CHK(x₁)) = [4]x₁   
POL(F(x₁)) = [2]x₁   
POL(TP(x₁)) = [4] + [4]x₁   
POL(X) = 0   
POL(active(x₁)) = [2] + [3]x₁   
POL(c) = 0   
POL(c1(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c2(x₁, x₂)) = x₁ + x₂   
POL(c6(x₁, x₂)) = x₁ + x₂   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(c9(x₁, x₂)) = x₁ + x₂   
POL(chk(x₁)) = [2]   
POL(f(x₁)) = [1] + [2]x₁   
POL(mark(x₁)) = [2] + x₁   
POL(mat(x₁, x₂)) = [2]x₂   
POL(no(x₁)) = x₁   
POL(y) = 0   

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(active(z0)) → active(f(z0))
f(no(z0)) → no(f(z0))
f(mark(z0)) → mark(f(z0))
active(f(z0)) → mark(f(f(z0)))
chk(no(c)) → active(c)
mat(f(z0), c) → no(c)
mat(f(z0), f(y)) → f(mat(z0, y))

Tuples:

F(active(z0)) → c6(ACTIVE(f(z0)), F(z0))
F(no(z0)) → c7(F(z0))
F(mark(z0)) → c8(F(z0))
ACTIVE(f(z0)) → c1(F(z0))
CHK(no(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))
CHK(no(f(f(y)))) → c2(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))
TP(mark(c)) → c9(TP(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))
TP(mark(f(y))) → c9(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))

S tuples:none
K tuples:

F(no(z0)) → c7(F(z0))
CHK(no(f(f(y)))) → c2(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))
TP(mark(f(y))) → c9(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))
TP(mark(c)) → c9(TP(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))
CHK(no(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)))
F(active(z0)) → c6(ACTIVE(f(z0)), F(z0))
F(mark(z0)) → c8(F(z0))
ACTIVE(f(z0)) → c1(F(z0))

Defined Rule Symbols:

f, active, chk, mat

Defined Pair Symbols:

F, ACTIVE, CHK, TP

Compound Symbols:

c6, c7, c8, c1, c2, c2, c9, c9

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

The set S is empty

(32) BOUNDS(O(1), O(1))