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

0 CpxTRS
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)
↳2 CdtProblem
    ↳3 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳4 CdtProblem
    ↳5 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳6 CdtProblem
    ↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 1257 ms)
    ↳8 CdtProblem
    ↳9 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 15 ms)
    ↳10 CdtProblem
    ↳11 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳12 CdtProblem
    ↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 727 ms)
    ↳14 CdtProblem
    ↳15 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳16 CdtProblem
    ↳17 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳18 CdtProblem
    ↳19 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 766 ms)
    ↳20 CdtProblem
    ↳21 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 2036 ms)
    ↳22 CdtProblem
    ↳23 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳24 CdtProblem
    ↳25 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳26 CdtProblem
    ↳27 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳28 CdtProblem
    ↳29 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 8 ms)
    ↳30 CdtProblem
↳31 CpxTrsMatchBoundsTAProof (⇔, 38 ms)
↳32 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 CpxTRS 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))
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))
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, c6, c7, c8, c9

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

Removed 2 trailing nodes:

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

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

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

ACTIVE(f(x0)) → c1(F(x0))

(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:

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))
ACTIVE(f(x0)) → c1(F(x0))
S tuples:

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))
ACTIVE(f(x0)) → c1(F(x0))
K tuples:none
Defined Rule Symbols:

active, chk, mat, f, tp

Defined Pair Symbols:

CHK, F, TP, ACTIVE

Compound Symbols:

c2, c6, c7, c8, c9, c1

(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.

F(no(z0)) → c7(F(z0))
We considered the (Usable) Rules:

mat(f(z0), f(y)) → f(mat(z0, y))
mat(f(z0), c) → no(c)
chk(no(f(z0))) → f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)))
chk(no(c)) → active(c)
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)))
And the Tuples:

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))
ACTIVE(f(x0)) → c1(F(x0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVE(x1)) = x1   
POL(CHK(x1)) = 0   
POL(F(x1)) = [4]x1   
POL(MAT(x1, x2)) = 0   
POL(TP(x1)) = 0   
POL(X) = 0   
POL(active(x1)) = [4]x1   
POL(c) = 0   
POL(c1(x1)) = x1   
POL(c2(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13)) = x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 + x12 + x13   
POL(c6(x1, x2)) = x1 + x2   
POL(c7(x1)) = x1   
POL(c8(x1)) = x1   
POL(c9(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13)) = x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 + x12 + x13   
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   

(8) 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:

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))
ACTIVE(f(x0)) → c1(F(x0))
S tuples:

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(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))
ACTIVE(f(x0)) → c1(F(x0))
K tuples:

F(no(z0)) → c7(F(z0))
Defined Rule Symbols:

active, chk, mat, f, tp

Defined Pair Symbols:

CHK, F, TP, ACTIVE

Compound Symbols:

c2, c6, c7, c8, c9, c1

(9) 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)), 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)) 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))), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y)), 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(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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(f(x0))) → c2

(10) 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))
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))
ACTIVE(f(x0)) → c1(F(x0))
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))), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y)), 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(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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(f(x0))) → c2
S tuples:

F(active(z0)) → c6(ACTIVE(f(z0)), 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))
ACTIVE(f(x0)) → c1(F(x0))
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))), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y)), 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(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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(f(x0))) → c2
K tuples:

F(no(z0)) → c7(F(z0))
Defined Rule Symbols:

active, chk, mat, f, tp

Defined Pair Symbols:

F, TP, ACTIVE, CHK

Compound Symbols:

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

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

Removed 1 trailing nodes:

CHK(no(f(x0))) → c2

(12) 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))
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))
ACTIVE(f(x0)) → c1(F(x0))
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))), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y)), 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(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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:

F(active(z0)) → c6(ACTIVE(f(z0)), 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))
ACTIVE(f(x0)) → c1(F(x0))
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))), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y)), 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(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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:

F(no(z0)) → c7(F(z0))
Defined Rule Symbols:

active, chk, mat, f, tp

Defined Pair Symbols:

F, TP, ACTIVE, CHK

Compound Symbols:

c6, c7, c8, c9, c1, c2

(13) 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.

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))), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y)), 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))
We considered the (Usable) Rules:

chk(no(c)) → active(c)
chk(no(f(z0))) → f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), z0)))
mat(f(z0), c) → no(c)
mat(f(z0), f(y)) → f(mat(z0, y))
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)))
And the Tuples:

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))
ACTIVE(f(x0)) → c1(F(x0))
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))), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y)), 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(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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))
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(MAT(x1, x2)) = 0   
POL(TP(x1)) = x1   
POL(X) = [3]   
POL(active(x1)) = x1   
POL(c) = 0   
POL(c1(x1)) = x1   
POL(c2(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13)) = x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 + x12 + x13   
POL(c6(x1, x2)) = x1 + x2   
POL(c7(x1)) = x1   
POL(c8(x1)) = x1   
POL(c9(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13)) = x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 + x12 + x13   
POL(chk(x1)) = 0   
POL(f(x1)) = x1   
POL(mark(x1)) = x1   
POL(mat(x1, x2)) = 0   
POL(no(x1)) = x1   
POL(y) = [1]   

(14) 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))
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))
ACTIVE(f(x0)) → c1(F(x0))
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))), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y)), 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(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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:

F(active(z0)) → c6(ACTIVE(f(z0)), 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))
ACTIVE(f(x0)) → c1(F(x0))
CHK(no(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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:

F(no(z0)) → c7(F(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))), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y)), 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))
Defined Rule Symbols:

active, chk, mat, f, tp

Defined Pair Symbols:

F, TP, ACTIVE, CHK

Compound Symbols:

c6, c7, c8, c9, c1, c2

(15) 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)), 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)) 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))), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y)), 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))
TP(mark(c)) → c9(TP(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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))
TP(mark(x0)) → c9

(16) 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(x0)) → c1(F(x0))
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))), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y)), 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(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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))
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))), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y)), 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))
TP(mark(c)) → c9(TP(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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))
TP(mark(x0)) → c9
S tuples:

F(active(z0)) → c6(ACTIVE(f(z0)), F(z0))
F(mark(z0)) → c8(F(z0))
ACTIVE(f(x0)) → c1(F(x0))
CHK(no(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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))
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))), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y)), 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))
TP(mark(c)) → c9(TP(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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))
TP(mark(x0)) → c9
K tuples:

F(no(z0)) → c7(F(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))), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y)), 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))
Defined Rule Symbols:

active, chk, mat, f, tp

Defined Pair Symbols:

F, ACTIVE, CHK, TP

Compound Symbols:

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

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

Removed 1 trailing nodes:

TP(mark(x0)) → c9

(18) 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(x0)) → c1(F(x0))
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))), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y)), 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(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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))
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))), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y)), 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))
TP(mark(c)) → c9(TP(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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:

F(active(z0)) → c6(ACTIVE(f(z0)), F(z0))
F(mark(z0)) → c8(F(z0))
ACTIVE(f(x0)) → c1(F(x0))
CHK(no(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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))
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))), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y)), 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))
TP(mark(c)) → c9(TP(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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:

F(no(z0)) → c7(F(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))), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y)), 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))
Defined Rule Symbols:

active, chk, mat, f, tp

Defined Pair Symbols:

F, ACTIVE, CHK, TP

Compound Symbols:

c6, c7, c8, c1, c2, c9

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

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

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))), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y)), 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))
TP(mark(c)) → c9(TP(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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))
We considered the (Usable) Rules:

chk(no(c)) → active(c)
mat(f(z0), c) → no(c)
mat(f(z0), f(y)) → f(mat(z0, y))
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)))
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(x0)) → c1(F(x0))
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))), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y)), 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(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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))
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))), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y)), 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))
TP(mark(c)) → c9(TP(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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))
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(MAT(x1, x2)) = 0   
POL(TP(x1)) = x1   
POL(X) = [2]   
POL(active(x1)) = 0   
POL(c) = 0   
POL(c1(x1)) = x1   
POL(c2(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13)) = x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 + x12 + x13   
POL(c6(x1, x2)) = x1 + x2   
POL(c7(x1)) = x1   
POL(c8(x1)) = x1   
POL(c9(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13)) = x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 + x12 + x13   
POL(chk(x1)) = 0   
POL(f(x1)) = 0   
POL(mark(x1)) = [1]   
POL(mat(x1, x2)) = 0   
POL(no(x1)) = 0   
POL(y) = 0   

(20) 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(x0)) → c1(F(x0))
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))), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y)), 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(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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))
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))), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y)), 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))
TP(mark(c)) → c9(TP(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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:

F(active(z0)) → c6(ACTIVE(f(z0)), F(z0))
F(mark(z0)) → c8(F(z0))
ACTIVE(f(x0)) → c1(F(x0))
CHK(no(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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:

F(no(z0)) → c7(F(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))), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y)), 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))
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))), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y)), 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))
TP(mark(c)) → c9(TP(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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))
Defined Rule Symbols:

active, chk, mat, f, tp

Defined Pair Symbols:

F, ACTIVE, CHK, TP

Compound Symbols:

c6, c7, c8, c1, c2, c9

(21) 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.

CHK(no(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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))
We considered the (Usable) Rules:

chk(no(c)) → active(c)
mat(f(z0), c) → no(c)
mat(f(z0), f(y)) → f(mat(z0, y))
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)))
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(x0)) → c1(F(x0))
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))), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y)), 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(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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))
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))), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y)), 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))
TP(mark(c)) → c9(TP(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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))
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(MAT(x1, x2)) = 0   
POL(TP(x1)) = x1   
POL(X) = 0   
POL(active(x1)) = [2]x1   
POL(c) = 0   
POL(c1(x1)) = x1   
POL(c2(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13)) = x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 + x12 + x13   
POL(c6(x1, x2)) = x1 + x2   
POL(c7(x1)) = x1   
POL(c8(x1)) = x1   
POL(c9(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13)) = x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 + x12 + x13   
POL(chk(x1)) = 0   
POL(f(x1)) = [1]   
POL(mark(x1)) = [1]   
POL(mat(x1, x2)) = x2   
POL(no(x1)) = x1   
POL(y) = 0   

(22) 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(x0)) → c1(F(x0))
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))), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y)), 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(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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))
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))), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y)), 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))
TP(mark(c)) → c9(TP(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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:

F(active(z0)) → c6(ACTIVE(f(z0)), F(z0))
F(mark(z0)) → c8(F(z0))
ACTIVE(f(x0)) → c1(F(x0))
K tuples:

F(no(z0)) → c7(F(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))), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y)), 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))
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))), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y)), 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))
TP(mark(c)) → c9(TP(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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))
Defined Rule Symbols:

active, chk, mat, f, tp

Defined Pair Symbols:

F, ACTIVE, CHK, TP

Compound Symbols:

c6, c7, c8, c1, c2, c9

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

Use narrowing to replace 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))), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y)), 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)) by

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

(24) 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(x0)) → c1(F(x0))
CHK(no(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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))
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))), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y)), 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))
TP(mark(c)) → c9(TP(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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(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(x0)) → c1(F(x0))
K tuples:

F(no(z0)) → c7(F(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))), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y)), 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))
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))), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y)), 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))
TP(mark(c)) → c9(TP(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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))
Defined Rule Symbols:

active, chk, mat, f, tp

Defined Pair Symbols:

F, ACTIVE, CHK, TP

Compound Symbols:

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

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

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

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

(26) 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(x0)) → c1(F(x0))
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))), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y)), 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))
TP(mark(c)) → c9(TP(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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(f(f(y)))) → c2(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))
CHK(no(f(c))) → c2(F(active(c)), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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(f(c))) → c2
S tuples:

F(active(z0)) → c6(ACTIVE(f(z0)), F(z0))
F(mark(z0)) → c8(F(z0))
ACTIVE(f(x0)) → c1(F(x0))
K tuples:

F(no(z0)) → c7(F(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))), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y)), 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))
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))), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y)), 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))
TP(mark(c)) → c9(TP(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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(f(c))) → c2(F(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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))
Defined Rule Symbols:

active, chk, mat, f, tp

Defined Pair Symbols:

F, ACTIVE, TP, CHK

Compound Symbols:

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

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

Removed 1 trailing nodes:

CHK(no(f(c))) → c2

(28) 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(x0)) → c1(F(x0))
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))), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y)), 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))
TP(mark(c)) → c9(TP(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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(f(f(y)))) → c2(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))
CHK(no(f(c))) → c2(F(active(c)), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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:

F(active(z0)) → c6(ACTIVE(f(z0)), F(z0))
F(mark(z0)) → c8(F(z0))
ACTIVE(f(x0)) → c1(F(x0))
K tuples:

F(no(z0)) → c7(F(z0))
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))), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y)), 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))
TP(mark(c)) → c9(TP(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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))
Defined Rule Symbols:

active, chk, mat, f, tp

Defined Pair Symbols:

F, ACTIVE, TP, CHK

Compound Symbols:

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

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

Use narrowing to replace 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))), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y)), 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)) by

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

(30) 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(x0)) → c1(F(x0))
TP(mark(c)) → c9(TP(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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(f(f(y)))) → c2(CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y))))
CHK(no(f(c))) → c2(F(active(c)), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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))
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(x0)) → c1(F(x0))
K tuples:

F(no(z0)) → c7(F(z0))
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))), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), f(y)), 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))
TP(mark(c)) → c9(TP(chk(no(c))), CHK(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))), c)), MAT(f(f(f(f(f(f(f(f(f(f(X)))))))))), c), 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))
Defined Rule Symbols:

active, chk, mat, f, tp

Defined Pair Symbols:

F, ACTIVE, TP, CHK

Compound Symbols:

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

(31) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)

A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 2.

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1, 2, 3, 4, 5]
transitions:
mark0(0) → 0
no0(0) → 0
X0() → 0
y0() → 0
c0() → 0
active0(0) → 1
chk0(0) → 2
mat0(0, 0) → 3
f0(0) → 4
tp0(0) → 5
c1() → 6
active1(6) → 2
f1(0) → 7
no1(7) → 4
f1(0) → 8
mark1(8) → 4
X1() → 14
f1(14) → 13
f1(13) → 12
f1(12) → 12
f1(12) → 11
mat1(11, 0) → 10
chk1(10) → 9
tp1(9) → 5
c1() → 15
no1(15) → 10
no1(7) → 7
no1(7) → 8
mark1(8) → 7
mark1(8) → 8
c2() → 16
active2(16) → 9

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