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 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 612 ms)
    ↳4 CdtProblem
    ↳5 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳6 CdtProblem
    ↳7 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳8 CdtProblem
    ↳9 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳10 CdtProblem
    ↳11 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳12 CdtProblem
    ↳13 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳14 CdtProblem
    ↳15 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳16 CdtProblem
    ↳17 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳18 CdtProblem
    ↳19 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳20 CdtProblem
    ↳21 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳22 CdtProblem
    ↳23 CdtLeafRemovalProof (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), 0 ms)
    ↳30 CdtProblem
    ↳31 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳32 CdtProblem
↳33 CpxTrsMatchBoundsTAProof (⇔, 196 ms)
↳34 BOUNDS(O(1), O(n^1))

(0) Obligation:

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

active(f(x)) → mark(x)
top(active(c)) → top(mark(c))
top(mark(x)) → top(check(x))
check(f(x)) → f(check(x))
check(x) → start(match(f(X), x))
match(f(x), f(y)) → f(match(x, y))
match(X, x) → proper(x)
proper(c) → ok(c)
proper(f(x)) → f(proper(x))
f(ok(x)) → ok(f(x))
start(ok(x)) → found(x)
f(found(x)) → found(f(x))
top(found(x)) → top(active(x))
active(f(x)) → f(active(x))
f(mark(x)) → mark(f(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(z0)
active(f(z0)) → f(active(z0))
top(active(c)) → top(mark(c))
top(mark(z0)) → top(check(z0))
top(found(z0)) → top(active(z0))
check(f(z0)) → f(check(z0))
check(z0) → start(match(f(X), z0))
match(f(z0), f(z1)) → f(match(z0, z1))
match(X, z0) → proper(z0)
proper(c) → ok(c)
proper(f(z0)) → f(proper(z0))
f(ok(z0)) → ok(f(z0))
f(found(z0)) → found(f(z0))
f(mark(z0)) → mark(f(z0))
start(ok(z0)) → found(z0)
Tuples:

ACTIVE(f(z0)) → c2(F(active(z0)), ACTIVE(z0))
TOP(active(c)) → c3(TOP(mark(c)))
TOP(mark(z0)) → c4(TOP(check(z0)), CHECK(z0))
TOP(found(z0)) → c5(TOP(active(z0)), ACTIVE(z0))
CHECK(f(z0)) → c6(F(check(z0)), CHECK(z0))
CHECK(z0) → c7(START(match(f(X), z0)), MATCH(f(X), z0), F(X))
MATCH(f(z0), f(z1)) → c8(F(match(z0, z1)), MATCH(z0, z1))
MATCH(X, z0) → c9(PROPER(z0))
PROPER(f(z0)) → c11(F(proper(z0)), PROPER(z0))
F(ok(z0)) → c12(F(z0))
F(found(z0)) → c13(F(z0))
F(mark(z0)) → c14(F(z0))
S tuples:

ACTIVE(f(z0)) → c2(F(active(z0)), ACTIVE(z0))
TOP(active(c)) → c3(TOP(mark(c)))
TOP(mark(z0)) → c4(TOP(check(z0)), CHECK(z0))
TOP(found(z0)) → c5(TOP(active(z0)), ACTIVE(z0))
CHECK(f(z0)) → c6(F(check(z0)), CHECK(z0))
CHECK(z0) → c7(START(match(f(X), z0)), MATCH(f(X), z0), F(X))
MATCH(f(z0), f(z1)) → c8(F(match(z0, z1)), MATCH(z0, z1))
MATCH(X, z0) → c9(PROPER(z0))
PROPER(f(z0)) → c11(F(proper(z0)), PROPER(z0))
F(ok(z0)) → c12(F(z0))
F(found(z0)) → c13(F(z0))
F(mark(z0)) → c14(F(z0))
K tuples:none
Defined Rule Symbols:

active, top, check, match, proper, f, start

Defined Pair Symbols:

ACTIVE, TOP, CHECK, MATCH, PROPER, F

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c11, c12, c13, c14

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

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

TOP(active(c)) → c3(TOP(mark(c)))
We considered the (Usable) Rules:

proper(c) → ok(c)
proper(f(z0)) → f(proper(z0))
f(ok(z0)) → ok(f(z0))
f(found(z0)) → found(f(z0))
f(mark(z0)) → mark(f(z0))
match(f(z0), f(z1)) → f(match(z0, z1))
match(X, z0) → proper(z0)
check(f(z0)) → f(check(z0))
check(z0) → start(match(f(X), z0))
start(ok(z0)) → found(z0)
active(f(z0)) → mark(z0)
active(f(z0)) → f(active(z0))
And the Tuples:

ACTIVE(f(z0)) → c2(F(active(z0)), ACTIVE(z0))
TOP(active(c)) → c3(TOP(mark(c)))
TOP(mark(z0)) → c4(TOP(check(z0)), CHECK(z0))
TOP(found(z0)) → c5(TOP(active(z0)), ACTIVE(z0))
CHECK(f(z0)) → c6(F(check(z0)), CHECK(z0))
CHECK(z0) → c7(START(match(f(X), z0)), MATCH(f(X), z0), F(X))
MATCH(f(z0), f(z1)) → c8(F(match(z0, z1)), MATCH(z0, z1))
MATCH(X, z0) → c9(PROPER(z0))
PROPER(f(z0)) → c11(F(proper(z0)), PROPER(z0))
F(ok(z0)) → c12(F(z0))
F(found(z0)) → c13(F(z0))
F(mark(z0)) → c14(F(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVE(x1)) = 0   
POL(CHECK(x1)) = 0   
POL(F(x1)) = 0   
POL(MATCH(x1, x2)) = 0   
POL(PROPER(x1)) = 0   
POL(START(x1)) = 0   
POL(TOP(x1)) = [2]x1   
POL(X) = 0   
POL(active(x1)) = x1   
POL(c) = [1]   
POL(c11(x1, x2)) = x1 + x2   
POL(c12(x1)) = x1   
POL(c13(x1)) = x1   
POL(c14(x1)) = x1   
POL(c2(x1, x2)) = x1 + x2   
POL(c3(x1)) = x1   
POL(c4(x1, x2)) = x1 + x2   
POL(c5(x1, x2)) = x1 + x2   
POL(c6(x1, x2)) = x1 + x2   
POL(c7(x1, x2, x3)) = x1 + x2 + x3   
POL(c8(x1, x2)) = x1 + x2   
POL(c9(x1)) = x1   
POL(check(x1)) = 0   
POL(f(x1)) = 0   
POL(found(x1)) = x1   
POL(mark(x1)) = 0   
POL(match(x1, x2)) = 0   
POL(ok(x1)) = x1   
POL(proper(x1)) = 0   
POL(start(x1)) = [4]x1   

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(z0)
active(f(z0)) → f(active(z0))
top(active(c)) → top(mark(c))
top(mark(z0)) → top(check(z0))
top(found(z0)) → top(active(z0))
check(f(z0)) → f(check(z0))
check(z0) → start(match(f(X), z0))
match(f(z0), f(z1)) → f(match(z0, z1))
match(X, z0) → proper(z0)
proper(c) → ok(c)
proper(f(z0)) → f(proper(z0))
f(ok(z0)) → ok(f(z0))
f(found(z0)) → found(f(z0))
f(mark(z0)) → mark(f(z0))
start(ok(z0)) → found(z0)
Tuples:

ACTIVE(f(z0)) → c2(F(active(z0)), ACTIVE(z0))
TOP(active(c)) → c3(TOP(mark(c)))
TOP(mark(z0)) → c4(TOP(check(z0)), CHECK(z0))
TOP(found(z0)) → c5(TOP(active(z0)), ACTIVE(z0))
CHECK(f(z0)) → c6(F(check(z0)), CHECK(z0))
CHECK(z0) → c7(START(match(f(X), z0)), MATCH(f(X), z0), F(X))
MATCH(f(z0), f(z1)) → c8(F(match(z0, z1)), MATCH(z0, z1))
MATCH(X, z0) → c9(PROPER(z0))
PROPER(f(z0)) → c11(F(proper(z0)), PROPER(z0))
F(ok(z0)) → c12(F(z0))
F(found(z0)) → c13(F(z0))
F(mark(z0)) → c14(F(z0))
S tuples:

ACTIVE(f(z0)) → c2(F(active(z0)), ACTIVE(z0))
TOP(mark(z0)) → c4(TOP(check(z0)), CHECK(z0))
TOP(found(z0)) → c5(TOP(active(z0)), ACTIVE(z0))
CHECK(f(z0)) → c6(F(check(z0)), CHECK(z0))
CHECK(z0) → c7(START(match(f(X), z0)), MATCH(f(X), z0), F(X))
MATCH(f(z0), f(z1)) → c8(F(match(z0, z1)), MATCH(z0, z1))
MATCH(X, z0) → c9(PROPER(z0))
PROPER(f(z0)) → c11(F(proper(z0)), PROPER(z0))
F(ok(z0)) → c12(F(z0))
F(found(z0)) → c13(F(z0))
F(mark(z0)) → c14(F(z0))
K tuples:

TOP(active(c)) → c3(TOP(mark(c)))
Defined Rule Symbols:

active, top, check, match, proper, f, start

Defined Pair Symbols:

ACTIVE, TOP, CHECK, MATCH, PROPER, F

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c11, c12, c13, c14

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

Use narrowing to replace ACTIVE(f(z0)) → c2(F(active(z0)), ACTIVE(z0)) by

ACTIVE(f(f(z0))) → c2(F(mark(z0)), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c2(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(x0)) → c2

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(z0)
active(f(z0)) → f(active(z0))
top(active(c)) → top(mark(c))
top(mark(z0)) → top(check(z0))
top(found(z0)) → top(active(z0))
check(f(z0)) → f(check(z0))
check(z0) → start(match(f(X), z0))
match(f(z0), f(z1)) → f(match(z0, z1))
match(X, z0) → proper(z0)
proper(c) → ok(c)
proper(f(z0)) → f(proper(z0))
f(ok(z0)) → ok(f(z0))
f(found(z0)) → found(f(z0))
f(mark(z0)) → mark(f(z0))
start(ok(z0)) → found(z0)
Tuples:

TOP(active(c)) → c3(TOP(mark(c)))
TOP(mark(z0)) → c4(TOP(check(z0)), CHECK(z0))
TOP(found(z0)) → c5(TOP(active(z0)), ACTIVE(z0))
CHECK(f(z0)) → c6(F(check(z0)), CHECK(z0))
CHECK(z0) → c7(START(match(f(X), z0)), MATCH(f(X), z0), F(X))
MATCH(f(z0), f(z1)) → c8(F(match(z0, z1)), MATCH(z0, z1))
MATCH(X, z0) → c9(PROPER(z0))
PROPER(f(z0)) → c11(F(proper(z0)), PROPER(z0))
F(ok(z0)) → c12(F(z0))
F(found(z0)) → c13(F(z0))
F(mark(z0)) → c14(F(z0))
ACTIVE(f(f(z0))) → c2(F(mark(z0)), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c2(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(x0)) → c2
S tuples:

TOP(mark(z0)) → c4(TOP(check(z0)), CHECK(z0))
TOP(found(z0)) → c5(TOP(active(z0)), ACTIVE(z0))
CHECK(f(z0)) → c6(F(check(z0)), CHECK(z0))
CHECK(z0) → c7(START(match(f(X), z0)), MATCH(f(X), z0), F(X))
MATCH(f(z0), f(z1)) → c8(F(match(z0, z1)), MATCH(z0, z1))
MATCH(X, z0) → c9(PROPER(z0))
PROPER(f(z0)) → c11(F(proper(z0)), PROPER(z0))
F(ok(z0)) → c12(F(z0))
F(found(z0)) → c13(F(z0))
F(mark(z0)) → c14(F(z0))
ACTIVE(f(f(z0))) → c2(F(mark(z0)), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c2(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(x0)) → c2
K tuples:

TOP(active(c)) → c3(TOP(mark(c)))
Defined Rule Symbols:

active, top, check, match, proper, f, start

Defined Pair Symbols:

TOP, CHECK, MATCH, PROPER, F, ACTIVE

Compound Symbols:

c3, c4, c5, c6, c7, c8, c9, c11, c12, c13, c14, c2, c2

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

Removed 1 trailing nodes:

ACTIVE(f(x0)) → c2

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(z0)
active(f(z0)) → f(active(z0))
top(active(c)) → top(mark(c))
top(mark(z0)) → top(check(z0))
top(found(z0)) → top(active(z0))
check(f(z0)) → f(check(z0))
check(z0) → start(match(f(X), z0))
match(f(z0), f(z1)) → f(match(z0, z1))
match(X, z0) → proper(z0)
proper(c) → ok(c)
proper(f(z0)) → f(proper(z0))
f(ok(z0)) → ok(f(z0))
f(found(z0)) → found(f(z0))
f(mark(z0)) → mark(f(z0))
start(ok(z0)) → found(z0)
Tuples:

TOP(active(c)) → c3(TOP(mark(c)))
TOP(mark(z0)) → c4(TOP(check(z0)), CHECK(z0))
TOP(found(z0)) → c5(TOP(active(z0)), ACTIVE(z0))
CHECK(f(z0)) → c6(F(check(z0)), CHECK(z0))
CHECK(z0) → c7(START(match(f(X), z0)), MATCH(f(X), z0), F(X))
MATCH(f(z0), f(z1)) → c8(F(match(z0, z1)), MATCH(z0, z1))
MATCH(X, z0) → c9(PROPER(z0))
PROPER(f(z0)) → c11(F(proper(z0)), PROPER(z0))
F(ok(z0)) → c12(F(z0))
F(found(z0)) → c13(F(z0))
F(mark(z0)) → c14(F(z0))
ACTIVE(f(f(z0))) → c2(F(mark(z0)), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c2(F(f(active(z0))), ACTIVE(f(z0)))
S tuples:

TOP(mark(z0)) → c4(TOP(check(z0)), CHECK(z0))
TOP(found(z0)) → c5(TOP(active(z0)), ACTIVE(z0))
CHECK(f(z0)) → c6(F(check(z0)), CHECK(z0))
CHECK(z0) → c7(START(match(f(X), z0)), MATCH(f(X), z0), F(X))
MATCH(f(z0), f(z1)) → c8(F(match(z0, z1)), MATCH(z0, z1))
MATCH(X, z0) → c9(PROPER(z0))
PROPER(f(z0)) → c11(F(proper(z0)), PROPER(z0))
F(ok(z0)) → c12(F(z0))
F(found(z0)) → c13(F(z0))
F(mark(z0)) → c14(F(z0))
ACTIVE(f(f(z0))) → c2(F(mark(z0)), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c2(F(f(active(z0))), ACTIVE(f(z0)))
K tuples:

TOP(active(c)) → c3(TOP(mark(c)))
Defined Rule Symbols:

active, top, check, match, proper, f, start

Defined Pair Symbols:

TOP, CHECK, MATCH, PROPER, F, ACTIVE

Compound Symbols:

c3, c4, c5, c6, c7, c8, c9, c11, c12, c13, c14, c2

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

Use narrowing to replace TOP(mark(z0)) → c4(TOP(check(z0)), CHECK(z0)) by

TOP(mark(f(z0))) → c4(TOP(f(check(z0))), CHECK(f(z0)))
TOP(mark(z0)) → c4(TOP(start(match(f(X), z0))), CHECK(z0))
TOP(mark(x0)) → c4

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(z0)
active(f(z0)) → f(active(z0))
top(active(c)) → top(mark(c))
top(mark(z0)) → top(check(z0))
top(found(z0)) → top(active(z0))
check(f(z0)) → f(check(z0))
check(z0) → start(match(f(X), z0))
match(f(z0), f(z1)) → f(match(z0, z1))
match(X, z0) → proper(z0)
proper(c) → ok(c)
proper(f(z0)) → f(proper(z0))
f(ok(z0)) → ok(f(z0))
f(found(z0)) → found(f(z0))
f(mark(z0)) → mark(f(z0))
start(ok(z0)) → found(z0)
Tuples:

TOP(active(c)) → c3(TOP(mark(c)))
TOP(found(z0)) → c5(TOP(active(z0)), ACTIVE(z0))
CHECK(f(z0)) → c6(F(check(z0)), CHECK(z0))
CHECK(z0) → c7(START(match(f(X), z0)), MATCH(f(X), z0), F(X))
MATCH(f(z0), f(z1)) → c8(F(match(z0, z1)), MATCH(z0, z1))
MATCH(X, z0) → c9(PROPER(z0))
PROPER(f(z0)) → c11(F(proper(z0)), PROPER(z0))
F(ok(z0)) → c12(F(z0))
F(found(z0)) → c13(F(z0))
F(mark(z0)) → c14(F(z0))
ACTIVE(f(f(z0))) → c2(F(mark(z0)), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c2(F(f(active(z0))), ACTIVE(f(z0)))
TOP(mark(f(z0))) → c4(TOP(f(check(z0))), CHECK(f(z0)))
TOP(mark(z0)) → c4(TOP(start(match(f(X), z0))), CHECK(z0))
TOP(mark(x0)) → c4
S tuples:

TOP(found(z0)) → c5(TOP(active(z0)), ACTIVE(z0))
CHECK(f(z0)) → c6(F(check(z0)), CHECK(z0))
CHECK(z0) → c7(START(match(f(X), z0)), MATCH(f(X), z0), F(X))
MATCH(f(z0), f(z1)) → c8(F(match(z0, z1)), MATCH(z0, z1))
MATCH(X, z0) → c9(PROPER(z0))
PROPER(f(z0)) → c11(F(proper(z0)), PROPER(z0))
F(ok(z0)) → c12(F(z0))
F(found(z0)) → c13(F(z0))
F(mark(z0)) → c14(F(z0))
ACTIVE(f(f(z0))) → c2(F(mark(z0)), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c2(F(f(active(z0))), ACTIVE(f(z0)))
TOP(mark(f(z0))) → c4(TOP(f(check(z0))), CHECK(f(z0)))
TOP(mark(z0)) → c4(TOP(start(match(f(X), z0))), CHECK(z0))
TOP(mark(x0)) → c4
K tuples:

TOP(active(c)) → c3(TOP(mark(c)))
Defined Rule Symbols:

active, top, check, match, proper, f, start

Defined Pair Symbols:

TOP, CHECK, MATCH, PROPER, F, ACTIVE

Compound Symbols:

c3, c5, c6, c7, c8, c9, c11, c12, c13, c14, c2, c4, c4

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

Removed 1 trailing nodes:

TOP(mark(x0)) → c4

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(z0)
active(f(z0)) → f(active(z0))
top(active(c)) → top(mark(c))
top(mark(z0)) → top(check(z0))
top(found(z0)) → top(active(z0))
check(f(z0)) → f(check(z0))
check(z0) → start(match(f(X), z0))
match(f(z0), f(z1)) → f(match(z0, z1))
match(X, z0) → proper(z0)
proper(c) → ok(c)
proper(f(z0)) → f(proper(z0))
f(ok(z0)) → ok(f(z0))
f(found(z0)) → found(f(z0))
f(mark(z0)) → mark(f(z0))
start(ok(z0)) → found(z0)
Tuples:

TOP(active(c)) → c3(TOP(mark(c)))
TOP(found(z0)) → c5(TOP(active(z0)), ACTIVE(z0))
CHECK(f(z0)) → c6(F(check(z0)), CHECK(z0))
CHECK(z0) → c7(START(match(f(X), z0)), MATCH(f(X), z0), F(X))
MATCH(f(z0), f(z1)) → c8(F(match(z0, z1)), MATCH(z0, z1))
MATCH(X, z0) → c9(PROPER(z0))
PROPER(f(z0)) → c11(F(proper(z0)), PROPER(z0))
F(ok(z0)) → c12(F(z0))
F(found(z0)) → c13(F(z0))
F(mark(z0)) → c14(F(z0))
ACTIVE(f(f(z0))) → c2(F(mark(z0)), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c2(F(f(active(z0))), ACTIVE(f(z0)))
TOP(mark(f(z0))) → c4(TOP(f(check(z0))), CHECK(f(z0)))
TOP(mark(z0)) → c4(TOP(start(match(f(X), z0))), CHECK(z0))
S tuples:

TOP(found(z0)) → c5(TOP(active(z0)), ACTIVE(z0))
CHECK(f(z0)) → c6(F(check(z0)), CHECK(z0))
CHECK(z0) → c7(START(match(f(X), z0)), MATCH(f(X), z0), F(X))
MATCH(f(z0), f(z1)) → c8(F(match(z0, z1)), MATCH(z0, z1))
MATCH(X, z0) → c9(PROPER(z0))
PROPER(f(z0)) → c11(F(proper(z0)), PROPER(z0))
F(ok(z0)) → c12(F(z0))
F(found(z0)) → c13(F(z0))
F(mark(z0)) → c14(F(z0))
ACTIVE(f(f(z0))) → c2(F(mark(z0)), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c2(F(f(active(z0))), ACTIVE(f(z0)))
TOP(mark(f(z0))) → c4(TOP(f(check(z0))), CHECK(f(z0)))
TOP(mark(z0)) → c4(TOP(start(match(f(X), z0))), CHECK(z0))
K tuples:

TOP(active(c)) → c3(TOP(mark(c)))
Defined Rule Symbols:

active, top, check, match, proper, f, start

Defined Pair Symbols:

TOP, CHECK, MATCH, PROPER, F, ACTIVE

Compound Symbols:

c3, c5, c6, c7, c8, c9, c11, c12, c13, c14, c2, c4

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

Use narrowing to replace CHECK(f(z0)) → c6(F(check(z0)), CHECK(z0)) by

CHECK(f(f(z0))) → c6(F(f(check(z0))), CHECK(f(z0)))
CHECK(f(z0)) → c6(F(start(match(f(X), z0))), CHECK(z0))
CHECK(f(x0)) → c6

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(z0)
active(f(z0)) → f(active(z0))
top(active(c)) → top(mark(c))
top(mark(z0)) → top(check(z0))
top(found(z0)) → top(active(z0))
check(f(z0)) → f(check(z0))
check(z0) → start(match(f(X), z0))
match(f(z0), f(z1)) → f(match(z0, z1))
match(X, z0) → proper(z0)
proper(c) → ok(c)
proper(f(z0)) → f(proper(z0))
f(ok(z0)) → ok(f(z0))
f(found(z0)) → found(f(z0))
f(mark(z0)) → mark(f(z0))
start(ok(z0)) → found(z0)
Tuples:

TOP(active(c)) → c3(TOP(mark(c)))
TOP(found(z0)) → c5(TOP(active(z0)), ACTIVE(z0))
CHECK(z0) → c7(START(match(f(X), z0)), MATCH(f(X), z0), F(X))
MATCH(f(z0), f(z1)) → c8(F(match(z0, z1)), MATCH(z0, z1))
MATCH(X, z0) → c9(PROPER(z0))
PROPER(f(z0)) → c11(F(proper(z0)), PROPER(z0))
F(ok(z0)) → c12(F(z0))
F(found(z0)) → c13(F(z0))
F(mark(z0)) → c14(F(z0))
ACTIVE(f(f(z0))) → c2(F(mark(z0)), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c2(F(f(active(z0))), ACTIVE(f(z0)))
TOP(mark(f(z0))) → c4(TOP(f(check(z0))), CHECK(f(z0)))
TOP(mark(z0)) → c4(TOP(start(match(f(X), z0))), CHECK(z0))
CHECK(f(f(z0))) → c6(F(f(check(z0))), CHECK(f(z0)))
CHECK(f(z0)) → c6(F(start(match(f(X), z0))), CHECK(z0))
CHECK(f(x0)) → c6
S tuples:

TOP(found(z0)) → c5(TOP(active(z0)), ACTIVE(z0))
CHECK(z0) → c7(START(match(f(X), z0)), MATCH(f(X), z0), F(X))
MATCH(f(z0), f(z1)) → c8(F(match(z0, z1)), MATCH(z0, z1))
MATCH(X, z0) → c9(PROPER(z0))
PROPER(f(z0)) → c11(F(proper(z0)), PROPER(z0))
F(ok(z0)) → c12(F(z0))
F(found(z0)) → c13(F(z0))
F(mark(z0)) → c14(F(z0))
ACTIVE(f(f(z0))) → c2(F(mark(z0)), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c2(F(f(active(z0))), ACTIVE(f(z0)))
TOP(mark(f(z0))) → c4(TOP(f(check(z0))), CHECK(f(z0)))
TOP(mark(z0)) → c4(TOP(start(match(f(X), z0))), CHECK(z0))
CHECK(f(f(z0))) → c6(F(f(check(z0))), CHECK(f(z0)))
CHECK(f(z0)) → c6(F(start(match(f(X), z0))), CHECK(z0))
CHECK(f(x0)) → c6
K tuples:

TOP(active(c)) → c3(TOP(mark(c)))
Defined Rule Symbols:

active, top, check, match, proper, f, start

Defined Pair Symbols:

TOP, CHECK, MATCH, PROPER, F, ACTIVE

Compound Symbols:

c3, c5, c7, c8, c9, c11, c12, c13, c14, c2, c4, c6, c6

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

Removed 1 trailing nodes:

CHECK(f(x0)) → c6

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(z0)
active(f(z0)) → f(active(z0))
top(active(c)) → top(mark(c))
top(mark(z0)) → top(check(z0))
top(found(z0)) → top(active(z0))
check(f(z0)) → f(check(z0))
check(z0) → start(match(f(X), z0))
match(f(z0), f(z1)) → f(match(z0, z1))
match(X, z0) → proper(z0)
proper(c) → ok(c)
proper(f(z0)) → f(proper(z0))
f(ok(z0)) → ok(f(z0))
f(found(z0)) → found(f(z0))
f(mark(z0)) → mark(f(z0))
start(ok(z0)) → found(z0)
Tuples:

TOP(active(c)) → c3(TOP(mark(c)))
TOP(found(z0)) → c5(TOP(active(z0)), ACTIVE(z0))
CHECK(z0) → c7(START(match(f(X), z0)), MATCH(f(X), z0), F(X))
MATCH(f(z0), f(z1)) → c8(F(match(z0, z1)), MATCH(z0, z1))
MATCH(X, z0) → c9(PROPER(z0))
PROPER(f(z0)) → c11(F(proper(z0)), PROPER(z0))
F(ok(z0)) → c12(F(z0))
F(found(z0)) → c13(F(z0))
F(mark(z0)) → c14(F(z0))
ACTIVE(f(f(z0))) → c2(F(mark(z0)), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c2(F(f(active(z0))), ACTIVE(f(z0)))
TOP(mark(f(z0))) → c4(TOP(f(check(z0))), CHECK(f(z0)))
TOP(mark(z0)) → c4(TOP(start(match(f(X), z0))), CHECK(z0))
CHECK(f(f(z0))) → c6(F(f(check(z0))), CHECK(f(z0)))
CHECK(f(z0)) → c6(F(start(match(f(X), z0))), CHECK(z0))
S tuples:

TOP(found(z0)) → c5(TOP(active(z0)), ACTIVE(z0))
CHECK(z0) → c7(START(match(f(X), z0)), MATCH(f(X), z0), F(X))
MATCH(f(z0), f(z1)) → c8(F(match(z0, z1)), MATCH(z0, z1))
MATCH(X, z0) → c9(PROPER(z0))
PROPER(f(z0)) → c11(F(proper(z0)), PROPER(z0))
F(ok(z0)) → c12(F(z0))
F(found(z0)) → c13(F(z0))
F(mark(z0)) → c14(F(z0))
ACTIVE(f(f(z0))) → c2(F(mark(z0)), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c2(F(f(active(z0))), ACTIVE(f(z0)))
TOP(mark(f(z0))) → c4(TOP(f(check(z0))), CHECK(f(z0)))
TOP(mark(z0)) → c4(TOP(start(match(f(X), z0))), CHECK(z0))
CHECK(f(f(z0))) → c6(F(f(check(z0))), CHECK(f(z0)))
CHECK(f(z0)) → c6(F(start(match(f(X), z0))), CHECK(z0))
K tuples:

TOP(active(c)) → c3(TOP(mark(c)))
Defined Rule Symbols:

active, top, check, match, proper, f, start

Defined Pair Symbols:

TOP, CHECK, MATCH, PROPER, F, ACTIVE

Compound Symbols:

c3, c5, c7, c8, c9, c11, c12, c13, c14, c2, c4, c6

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

Use narrowing to replace CHECK(z0) → c7(START(match(f(X), z0)), MATCH(f(X), z0), F(X)) by

CHECK(f(z1)) → c7(START(f(match(X, z1))), MATCH(f(X), f(z1)), F(X))
CHECK(x0) → c7

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(z0)
active(f(z0)) → f(active(z0))
top(active(c)) → top(mark(c))
top(mark(z0)) → top(check(z0))
top(found(z0)) → top(active(z0))
check(f(z0)) → f(check(z0))
check(z0) → start(match(f(X), z0))
match(f(z0), f(z1)) → f(match(z0, z1))
match(X, z0) → proper(z0)
proper(c) → ok(c)
proper(f(z0)) → f(proper(z0))
f(ok(z0)) → ok(f(z0))
f(found(z0)) → found(f(z0))
f(mark(z0)) → mark(f(z0))
start(ok(z0)) → found(z0)
Tuples:

TOP(active(c)) → c3(TOP(mark(c)))
TOP(found(z0)) → c5(TOP(active(z0)), ACTIVE(z0))
MATCH(f(z0), f(z1)) → c8(F(match(z0, z1)), MATCH(z0, z1))
MATCH(X, z0) → c9(PROPER(z0))
PROPER(f(z0)) → c11(F(proper(z0)), PROPER(z0))
F(ok(z0)) → c12(F(z0))
F(found(z0)) → c13(F(z0))
F(mark(z0)) → c14(F(z0))
ACTIVE(f(f(z0))) → c2(F(mark(z0)), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c2(F(f(active(z0))), ACTIVE(f(z0)))
TOP(mark(f(z0))) → c4(TOP(f(check(z0))), CHECK(f(z0)))
TOP(mark(z0)) → c4(TOP(start(match(f(X), z0))), CHECK(z0))
CHECK(f(f(z0))) → c6(F(f(check(z0))), CHECK(f(z0)))
CHECK(f(z0)) → c6(F(start(match(f(X), z0))), CHECK(z0))
CHECK(f(z1)) → c7(START(f(match(X, z1))), MATCH(f(X), f(z1)), F(X))
CHECK(x0) → c7
S tuples:

TOP(found(z0)) → c5(TOP(active(z0)), ACTIVE(z0))
MATCH(f(z0), f(z1)) → c8(F(match(z0, z1)), MATCH(z0, z1))
MATCH(X, z0) → c9(PROPER(z0))
PROPER(f(z0)) → c11(F(proper(z0)), PROPER(z0))
F(ok(z0)) → c12(F(z0))
F(found(z0)) → c13(F(z0))
F(mark(z0)) → c14(F(z0))
ACTIVE(f(f(z0))) → c2(F(mark(z0)), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c2(F(f(active(z0))), ACTIVE(f(z0)))
TOP(mark(f(z0))) → c4(TOP(f(check(z0))), CHECK(f(z0)))
TOP(mark(z0)) → c4(TOP(start(match(f(X), z0))), CHECK(z0))
CHECK(f(f(z0))) → c6(F(f(check(z0))), CHECK(f(z0)))
CHECK(f(z0)) → c6(F(start(match(f(X), z0))), CHECK(z0))
CHECK(f(z1)) → c7(START(f(match(X, z1))), MATCH(f(X), f(z1)), F(X))
CHECK(x0) → c7
K tuples:

TOP(active(c)) → c3(TOP(mark(c)))
Defined Rule Symbols:

active, top, check, match, proper, f, start

Defined Pair Symbols:

TOP, MATCH, PROPER, F, ACTIVE, CHECK

Compound Symbols:

c3, c5, c8, c9, c11, c12, c13, c14, c2, c4, c6, c7, c7

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

Removed 1 trailing nodes:

CHECK(x0) → c7

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(z0)
active(f(z0)) → f(active(z0))
top(active(c)) → top(mark(c))
top(mark(z0)) → top(check(z0))
top(found(z0)) → top(active(z0))
check(f(z0)) → f(check(z0))
check(z0) → start(match(f(X), z0))
match(f(z0), f(z1)) → f(match(z0, z1))
match(X, z0) → proper(z0)
proper(c) → ok(c)
proper(f(z0)) → f(proper(z0))
f(ok(z0)) → ok(f(z0))
f(found(z0)) → found(f(z0))
f(mark(z0)) → mark(f(z0))
start(ok(z0)) → found(z0)
Tuples:

TOP(active(c)) → c3(TOP(mark(c)))
TOP(found(z0)) → c5(TOP(active(z0)), ACTIVE(z0))
MATCH(f(z0), f(z1)) → c8(F(match(z0, z1)), MATCH(z0, z1))
MATCH(X, z0) → c9(PROPER(z0))
PROPER(f(z0)) → c11(F(proper(z0)), PROPER(z0))
F(ok(z0)) → c12(F(z0))
F(found(z0)) → c13(F(z0))
F(mark(z0)) → c14(F(z0))
ACTIVE(f(f(z0))) → c2(F(mark(z0)), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c2(F(f(active(z0))), ACTIVE(f(z0)))
TOP(mark(f(z0))) → c4(TOP(f(check(z0))), CHECK(f(z0)))
TOP(mark(z0)) → c4(TOP(start(match(f(X), z0))), CHECK(z0))
CHECK(f(f(z0))) → c6(F(f(check(z0))), CHECK(f(z0)))
CHECK(f(z0)) → c6(F(start(match(f(X), z0))), CHECK(z0))
CHECK(f(z1)) → c7(START(f(match(X, z1))), MATCH(f(X), f(z1)), F(X))
S tuples:

TOP(found(z0)) → c5(TOP(active(z0)), ACTIVE(z0))
MATCH(f(z0), f(z1)) → c8(F(match(z0, z1)), MATCH(z0, z1))
MATCH(X, z0) → c9(PROPER(z0))
PROPER(f(z0)) → c11(F(proper(z0)), PROPER(z0))
F(ok(z0)) → c12(F(z0))
F(found(z0)) → c13(F(z0))
F(mark(z0)) → c14(F(z0))
ACTIVE(f(f(z0))) → c2(F(mark(z0)), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c2(F(f(active(z0))), ACTIVE(f(z0)))
TOP(mark(f(z0))) → c4(TOP(f(check(z0))), CHECK(f(z0)))
TOP(mark(z0)) → c4(TOP(start(match(f(X), z0))), CHECK(z0))
CHECK(f(f(z0))) → c6(F(f(check(z0))), CHECK(f(z0)))
CHECK(f(z0)) → c6(F(start(match(f(X), z0))), CHECK(z0))
CHECK(f(z1)) → c7(START(f(match(X, z1))), MATCH(f(X), f(z1)), F(X))
K tuples:

TOP(active(c)) → c3(TOP(mark(c)))
Defined Rule Symbols:

active, top, check, match, proper, f, start

Defined Pair Symbols:

TOP, MATCH, PROPER, F, ACTIVE, CHECK

Compound Symbols:

c3, c5, c8, c9, c11, c12, c13, c14, c2, c4, c6, c7

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

Use narrowing to replace MATCH(f(z0), f(z1)) → c8(F(match(z0, z1)), MATCH(z0, z1)) by

MATCH(f(f(z0)), f(f(z1))) → c8(F(f(match(z0, z1))), MATCH(f(z0), f(z1)))
MATCH(f(X), f(z0)) → c8(F(proper(z0)), MATCH(X, z0))
MATCH(f(x0), f(x1)) → c8

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(z0)
active(f(z0)) → f(active(z0))
top(active(c)) → top(mark(c))
top(mark(z0)) → top(check(z0))
top(found(z0)) → top(active(z0))
check(f(z0)) → f(check(z0))
check(z0) → start(match(f(X), z0))
match(f(z0), f(z1)) → f(match(z0, z1))
match(X, z0) → proper(z0)
proper(c) → ok(c)
proper(f(z0)) → f(proper(z0))
f(ok(z0)) → ok(f(z0))
f(found(z0)) → found(f(z0))
f(mark(z0)) → mark(f(z0))
start(ok(z0)) → found(z0)
Tuples:

TOP(active(c)) → c3(TOP(mark(c)))
TOP(found(z0)) → c5(TOP(active(z0)), ACTIVE(z0))
MATCH(X, z0) → c9(PROPER(z0))
PROPER(f(z0)) → c11(F(proper(z0)), PROPER(z0))
F(ok(z0)) → c12(F(z0))
F(found(z0)) → c13(F(z0))
F(mark(z0)) → c14(F(z0))
ACTIVE(f(f(z0))) → c2(F(mark(z0)), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c2(F(f(active(z0))), ACTIVE(f(z0)))
TOP(mark(f(z0))) → c4(TOP(f(check(z0))), CHECK(f(z0)))
TOP(mark(z0)) → c4(TOP(start(match(f(X), z0))), CHECK(z0))
CHECK(f(f(z0))) → c6(F(f(check(z0))), CHECK(f(z0)))
CHECK(f(z0)) → c6(F(start(match(f(X), z0))), CHECK(z0))
CHECK(f(z1)) → c7(START(f(match(X, z1))), MATCH(f(X), f(z1)), F(X))
MATCH(f(f(z0)), f(f(z1))) → c8(F(f(match(z0, z1))), MATCH(f(z0), f(z1)))
MATCH(f(X), f(z0)) → c8(F(proper(z0)), MATCH(X, z0))
MATCH(f(x0), f(x1)) → c8
S tuples:

TOP(found(z0)) → c5(TOP(active(z0)), ACTIVE(z0))
MATCH(X, z0) → c9(PROPER(z0))
PROPER(f(z0)) → c11(F(proper(z0)), PROPER(z0))
F(ok(z0)) → c12(F(z0))
F(found(z0)) → c13(F(z0))
F(mark(z0)) → c14(F(z0))
ACTIVE(f(f(z0))) → c2(F(mark(z0)), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c2(F(f(active(z0))), ACTIVE(f(z0)))
TOP(mark(f(z0))) → c4(TOP(f(check(z0))), CHECK(f(z0)))
TOP(mark(z0)) → c4(TOP(start(match(f(X), z0))), CHECK(z0))
CHECK(f(f(z0))) → c6(F(f(check(z0))), CHECK(f(z0)))
CHECK(f(z0)) → c6(F(start(match(f(X), z0))), CHECK(z0))
CHECK(f(z1)) → c7(START(f(match(X, z1))), MATCH(f(X), f(z1)), F(X))
MATCH(f(f(z0)), f(f(z1))) → c8(F(f(match(z0, z1))), MATCH(f(z0), f(z1)))
MATCH(f(X), f(z0)) → c8(F(proper(z0)), MATCH(X, z0))
MATCH(f(x0), f(x1)) → c8
K tuples:

TOP(active(c)) → c3(TOP(mark(c)))
Defined Rule Symbols:

active, top, check, match, proper, f, start

Defined Pair Symbols:

TOP, MATCH, PROPER, F, ACTIVE, CHECK

Compound Symbols:

c3, c5, c9, c11, c12, c13, c14, c2, c4, c6, c7, c8, c8

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

Removed 1 trailing nodes:

MATCH(f(x0), f(x1)) → c8

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(z0)
active(f(z0)) → f(active(z0))
top(active(c)) → top(mark(c))
top(mark(z0)) → top(check(z0))
top(found(z0)) → top(active(z0))
check(f(z0)) → f(check(z0))
check(z0) → start(match(f(X), z0))
match(f(z0), f(z1)) → f(match(z0, z1))
match(X, z0) → proper(z0)
proper(c) → ok(c)
proper(f(z0)) → f(proper(z0))
f(ok(z0)) → ok(f(z0))
f(found(z0)) → found(f(z0))
f(mark(z0)) → mark(f(z0))
start(ok(z0)) → found(z0)
Tuples:

TOP(active(c)) → c3(TOP(mark(c)))
TOP(found(z0)) → c5(TOP(active(z0)), ACTIVE(z0))
MATCH(X, z0) → c9(PROPER(z0))
PROPER(f(z0)) → c11(F(proper(z0)), PROPER(z0))
F(ok(z0)) → c12(F(z0))
F(found(z0)) → c13(F(z0))
F(mark(z0)) → c14(F(z0))
ACTIVE(f(f(z0))) → c2(F(mark(z0)), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c2(F(f(active(z0))), ACTIVE(f(z0)))
TOP(mark(f(z0))) → c4(TOP(f(check(z0))), CHECK(f(z0)))
TOP(mark(z0)) → c4(TOP(start(match(f(X), z0))), CHECK(z0))
CHECK(f(f(z0))) → c6(F(f(check(z0))), CHECK(f(z0)))
CHECK(f(z0)) → c6(F(start(match(f(X), z0))), CHECK(z0))
CHECK(f(z1)) → c7(START(f(match(X, z1))), MATCH(f(X), f(z1)), F(X))
MATCH(f(f(z0)), f(f(z1))) → c8(F(f(match(z0, z1))), MATCH(f(z0), f(z1)))
MATCH(f(X), f(z0)) → c8(F(proper(z0)), MATCH(X, z0))
S tuples:

TOP(found(z0)) → c5(TOP(active(z0)), ACTIVE(z0))
MATCH(X, z0) → c9(PROPER(z0))
PROPER(f(z0)) → c11(F(proper(z0)), PROPER(z0))
F(ok(z0)) → c12(F(z0))
F(found(z0)) → c13(F(z0))
F(mark(z0)) → c14(F(z0))
ACTIVE(f(f(z0))) → c2(F(mark(z0)), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c2(F(f(active(z0))), ACTIVE(f(z0)))
TOP(mark(f(z0))) → c4(TOP(f(check(z0))), CHECK(f(z0)))
TOP(mark(z0)) → c4(TOP(start(match(f(X), z0))), CHECK(z0))
CHECK(f(f(z0))) → c6(F(f(check(z0))), CHECK(f(z0)))
CHECK(f(z0)) → c6(F(start(match(f(X), z0))), CHECK(z0))
CHECK(f(z1)) → c7(START(f(match(X, z1))), MATCH(f(X), f(z1)), F(X))
MATCH(f(f(z0)), f(f(z1))) → c8(F(f(match(z0, z1))), MATCH(f(z0), f(z1)))
MATCH(f(X), f(z0)) → c8(F(proper(z0)), MATCH(X, z0))
K tuples:

TOP(active(c)) → c3(TOP(mark(c)))
Defined Rule Symbols:

active, top, check, match, proper, f, start

Defined Pair Symbols:

TOP, MATCH, PROPER, F, ACTIVE, CHECK

Compound Symbols:

c3, c5, c9, c11, c12, c13, c14, c2, c4, c6, c7, c8

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

Use narrowing to replace PROPER(f(z0)) → c11(F(proper(z0)), PROPER(z0)) by

PROPER(f(c)) → c11(F(ok(c)), PROPER(c))
PROPER(f(f(z0))) → c11(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(x0)) → c11

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(z0)
active(f(z0)) → f(active(z0))
top(active(c)) → top(mark(c))
top(mark(z0)) → top(check(z0))
top(found(z0)) → top(active(z0))
check(f(z0)) → f(check(z0))
check(z0) → start(match(f(X), z0))
match(f(z0), f(z1)) → f(match(z0, z1))
match(X, z0) → proper(z0)
proper(c) → ok(c)
proper(f(z0)) → f(proper(z0))
f(ok(z0)) → ok(f(z0))
f(found(z0)) → found(f(z0))
f(mark(z0)) → mark(f(z0))
start(ok(z0)) → found(z0)
Tuples:

TOP(active(c)) → c3(TOP(mark(c)))
TOP(found(z0)) → c5(TOP(active(z0)), ACTIVE(z0))
MATCH(X, z0) → c9(PROPER(z0))
F(ok(z0)) → c12(F(z0))
F(found(z0)) → c13(F(z0))
F(mark(z0)) → c14(F(z0))
ACTIVE(f(f(z0))) → c2(F(mark(z0)), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c2(F(f(active(z0))), ACTIVE(f(z0)))
TOP(mark(f(z0))) → c4(TOP(f(check(z0))), CHECK(f(z0)))
TOP(mark(z0)) → c4(TOP(start(match(f(X), z0))), CHECK(z0))
CHECK(f(f(z0))) → c6(F(f(check(z0))), CHECK(f(z0)))
CHECK(f(z0)) → c6(F(start(match(f(X), z0))), CHECK(z0))
CHECK(f(z1)) → c7(START(f(match(X, z1))), MATCH(f(X), f(z1)), F(X))
MATCH(f(f(z0)), f(f(z1))) → c8(F(f(match(z0, z1))), MATCH(f(z0), f(z1)))
MATCH(f(X), f(z0)) → c8(F(proper(z0)), MATCH(X, z0))
PROPER(f(c)) → c11(F(ok(c)), PROPER(c))
PROPER(f(f(z0))) → c11(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(x0)) → c11
S tuples:

TOP(found(z0)) → c5(TOP(active(z0)), ACTIVE(z0))
MATCH(X, z0) → c9(PROPER(z0))
F(ok(z0)) → c12(F(z0))
F(found(z0)) → c13(F(z0))
F(mark(z0)) → c14(F(z0))
ACTIVE(f(f(z0))) → c2(F(mark(z0)), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c2(F(f(active(z0))), ACTIVE(f(z0)))
TOP(mark(f(z0))) → c4(TOP(f(check(z0))), CHECK(f(z0)))
TOP(mark(z0)) → c4(TOP(start(match(f(X), z0))), CHECK(z0))
CHECK(f(f(z0))) → c6(F(f(check(z0))), CHECK(f(z0)))
CHECK(f(z0)) → c6(F(start(match(f(X), z0))), CHECK(z0))
CHECK(f(z1)) → c7(START(f(match(X, z1))), MATCH(f(X), f(z1)), F(X))
MATCH(f(f(z0)), f(f(z1))) → c8(F(f(match(z0, z1))), MATCH(f(z0), f(z1)))
MATCH(f(X), f(z0)) → c8(F(proper(z0)), MATCH(X, z0))
PROPER(f(c)) → c11(F(ok(c)), PROPER(c))
PROPER(f(f(z0))) → c11(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(x0)) → c11
K tuples:

TOP(active(c)) → c3(TOP(mark(c)))
Defined Rule Symbols:

active, top, check, match, proper, f, start

Defined Pair Symbols:

TOP, MATCH, F, ACTIVE, CHECK, PROPER

Compound Symbols:

c3, c5, c9, c12, c13, c14, c2, c4, c6, c7, c8, c11, c11

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

Removed 1 trailing nodes:

PROPER(f(x0)) → c11

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(z0)
active(f(z0)) → f(active(z0))
top(active(c)) → top(mark(c))
top(mark(z0)) → top(check(z0))
top(found(z0)) → top(active(z0))
check(f(z0)) → f(check(z0))
check(z0) → start(match(f(X), z0))
match(f(z0), f(z1)) → f(match(z0, z1))
match(X, z0) → proper(z0)
proper(c) → ok(c)
proper(f(z0)) → f(proper(z0))
f(ok(z0)) → ok(f(z0))
f(found(z0)) → found(f(z0))
f(mark(z0)) → mark(f(z0))
start(ok(z0)) → found(z0)
Tuples:

TOP(active(c)) → c3(TOP(mark(c)))
TOP(found(z0)) → c5(TOP(active(z0)), ACTIVE(z0))
MATCH(X, z0) → c9(PROPER(z0))
F(ok(z0)) → c12(F(z0))
F(found(z0)) → c13(F(z0))
F(mark(z0)) → c14(F(z0))
ACTIVE(f(f(z0))) → c2(F(mark(z0)), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c2(F(f(active(z0))), ACTIVE(f(z0)))
TOP(mark(f(z0))) → c4(TOP(f(check(z0))), CHECK(f(z0)))
TOP(mark(z0)) → c4(TOP(start(match(f(X), z0))), CHECK(z0))
CHECK(f(f(z0))) → c6(F(f(check(z0))), CHECK(f(z0)))
CHECK(f(z0)) → c6(F(start(match(f(X), z0))), CHECK(z0))
CHECK(f(z1)) → c7(START(f(match(X, z1))), MATCH(f(X), f(z1)), F(X))
MATCH(f(f(z0)), f(f(z1))) → c8(F(f(match(z0, z1))), MATCH(f(z0), f(z1)))
MATCH(f(X), f(z0)) → c8(F(proper(z0)), MATCH(X, z0))
PROPER(f(c)) → c11(F(ok(c)), PROPER(c))
PROPER(f(f(z0))) → c11(F(f(proper(z0))), PROPER(f(z0)))
S tuples:

TOP(found(z0)) → c5(TOP(active(z0)), ACTIVE(z0))
MATCH(X, z0) → c9(PROPER(z0))
F(ok(z0)) → c12(F(z0))
F(found(z0)) → c13(F(z0))
F(mark(z0)) → c14(F(z0))
ACTIVE(f(f(z0))) → c2(F(mark(z0)), ACTIVE(f(z0)))
ACTIVE(f(f(z0))) → c2(F(f(active(z0))), ACTIVE(f(z0)))
TOP(mark(f(z0))) → c4(TOP(f(check(z0))), CHECK(f(z0)))
TOP(mark(z0)) → c4(TOP(start(match(f(X), z0))), CHECK(z0))
CHECK(f(f(z0))) → c6(F(f(check(z0))), CHECK(f(z0)))
CHECK(f(z0)) → c6(F(start(match(f(X), z0))), CHECK(z0))
CHECK(f(z1)) → c7(START(f(match(X, z1))), MATCH(f(X), f(z1)), F(X))
MATCH(f(f(z0)), f(f(z1))) → c8(F(f(match(z0, z1))), MATCH(f(z0), f(z1)))
MATCH(f(X), f(z0)) → c8(F(proper(z0)), MATCH(X, z0))
PROPER(f(c)) → c11(F(ok(c)), PROPER(c))
PROPER(f(f(z0))) → c11(F(f(proper(z0))), PROPER(f(z0)))
K tuples:

TOP(active(c)) → c3(TOP(mark(c)))
Defined Rule Symbols:

active, top, check, match, proper, f, start

Defined Pair Symbols:

TOP, MATCH, F, ACTIVE, CHECK, PROPER

Compound Symbols:

c3, c5, c9, c12, c13, c14, c2, c4, c6, c7, c8, c11

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

Use narrowing to replace ACTIVE(f(f(z0))) → c2(F(f(active(z0))), ACTIVE(f(z0))) by

ACTIVE(f(f(f(z0)))) → c2(F(f(mark(z0))), ACTIVE(f(f(z0))))
ACTIVE(f(f(f(z0)))) → c2(F(f(f(active(z0)))), ACTIVE(f(f(z0))))
ACTIVE(f(f(x0))) → c2

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(z0)
active(f(z0)) → f(active(z0))
top(active(c)) → top(mark(c))
top(mark(z0)) → top(check(z0))
top(found(z0)) → top(active(z0))
check(f(z0)) → f(check(z0))
check(z0) → start(match(f(X), z0))
match(f(z0), f(z1)) → f(match(z0, z1))
match(X, z0) → proper(z0)
proper(c) → ok(c)
proper(f(z0)) → f(proper(z0))
f(ok(z0)) → ok(f(z0))
f(found(z0)) → found(f(z0))
f(mark(z0)) → mark(f(z0))
start(ok(z0)) → found(z0)
Tuples:

TOP(active(c)) → c3(TOP(mark(c)))
TOP(found(z0)) → c5(TOP(active(z0)), ACTIVE(z0))
MATCH(X, z0) → c9(PROPER(z0))
F(ok(z0)) → c12(F(z0))
F(found(z0)) → c13(F(z0))
F(mark(z0)) → c14(F(z0))
ACTIVE(f(f(z0))) → c2(F(mark(z0)), ACTIVE(f(z0)))
TOP(mark(f(z0))) → c4(TOP(f(check(z0))), CHECK(f(z0)))
TOP(mark(z0)) → c4(TOP(start(match(f(X), z0))), CHECK(z0))
CHECK(f(f(z0))) → c6(F(f(check(z0))), CHECK(f(z0)))
CHECK(f(z0)) → c6(F(start(match(f(X), z0))), CHECK(z0))
CHECK(f(z1)) → c7(START(f(match(X, z1))), MATCH(f(X), f(z1)), F(X))
MATCH(f(f(z0)), f(f(z1))) → c8(F(f(match(z0, z1))), MATCH(f(z0), f(z1)))
MATCH(f(X), f(z0)) → c8(F(proper(z0)), MATCH(X, z0))
PROPER(f(c)) → c11(F(ok(c)), PROPER(c))
PROPER(f(f(z0))) → c11(F(f(proper(z0))), PROPER(f(z0)))
ACTIVE(f(f(f(z0)))) → c2(F(f(mark(z0))), ACTIVE(f(f(z0))))
ACTIVE(f(f(f(z0)))) → c2(F(f(f(active(z0)))), ACTIVE(f(f(z0))))
ACTIVE(f(f(x0))) → c2
S tuples:

TOP(found(z0)) → c5(TOP(active(z0)), ACTIVE(z0))
MATCH(X, z0) → c9(PROPER(z0))
F(ok(z0)) → c12(F(z0))
F(found(z0)) → c13(F(z0))
F(mark(z0)) → c14(F(z0))
ACTIVE(f(f(z0))) → c2(F(mark(z0)), ACTIVE(f(z0)))
TOP(mark(f(z0))) → c4(TOP(f(check(z0))), CHECK(f(z0)))
TOP(mark(z0)) → c4(TOP(start(match(f(X), z0))), CHECK(z0))
CHECK(f(f(z0))) → c6(F(f(check(z0))), CHECK(f(z0)))
CHECK(f(z0)) → c6(F(start(match(f(X), z0))), CHECK(z0))
CHECK(f(z1)) → c7(START(f(match(X, z1))), MATCH(f(X), f(z1)), F(X))
MATCH(f(f(z0)), f(f(z1))) → c8(F(f(match(z0, z1))), MATCH(f(z0), f(z1)))
MATCH(f(X), f(z0)) → c8(F(proper(z0)), MATCH(X, z0))
PROPER(f(c)) → c11(F(ok(c)), PROPER(c))
PROPER(f(f(z0))) → c11(F(f(proper(z0))), PROPER(f(z0)))
ACTIVE(f(f(f(z0)))) → c2(F(f(mark(z0))), ACTIVE(f(f(z0))))
ACTIVE(f(f(f(z0)))) → c2(F(f(f(active(z0)))), ACTIVE(f(f(z0))))
ACTIVE(f(f(x0))) → c2
K tuples:

TOP(active(c)) → c3(TOP(mark(c)))
Defined Rule Symbols:

active, top, check, match, proper, f, start

Defined Pair Symbols:

TOP, MATCH, F, ACTIVE, CHECK, PROPER

Compound Symbols:

c3, c5, c9, c12, c13, c14, c2, c4, c6, c7, c8, c11, c2

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

Removed 1 trailing nodes:

ACTIVE(f(f(x0))) → c2

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(z0)
active(f(z0)) → f(active(z0))
top(active(c)) → top(mark(c))
top(mark(z0)) → top(check(z0))
top(found(z0)) → top(active(z0))
check(f(z0)) → f(check(z0))
check(z0) → start(match(f(X), z0))
match(f(z0), f(z1)) → f(match(z0, z1))
match(X, z0) → proper(z0)
proper(c) → ok(c)
proper(f(z0)) → f(proper(z0))
f(ok(z0)) → ok(f(z0))
f(found(z0)) → found(f(z0))
f(mark(z0)) → mark(f(z0))
start(ok(z0)) → found(z0)
Tuples:

TOP(active(c)) → c3(TOP(mark(c)))
TOP(found(z0)) → c5(TOP(active(z0)), ACTIVE(z0))
MATCH(X, z0) → c9(PROPER(z0))
F(ok(z0)) → c12(F(z0))
F(found(z0)) → c13(F(z0))
F(mark(z0)) → c14(F(z0))
ACTIVE(f(f(z0))) → c2(F(mark(z0)), ACTIVE(f(z0)))
TOP(mark(f(z0))) → c4(TOP(f(check(z0))), CHECK(f(z0)))
TOP(mark(z0)) → c4(TOP(start(match(f(X), z0))), CHECK(z0))
CHECK(f(f(z0))) → c6(F(f(check(z0))), CHECK(f(z0)))
CHECK(f(z0)) → c6(F(start(match(f(X), z0))), CHECK(z0))
CHECK(f(z1)) → c7(START(f(match(X, z1))), MATCH(f(X), f(z1)), F(X))
MATCH(f(f(z0)), f(f(z1))) → c8(F(f(match(z0, z1))), MATCH(f(z0), f(z1)))
MATCH(f(X), f(z0)) → c8(F(proper(z0)), MATCH(X, z0))
PROPER(f(c)) → c11(F(ok(c)), PROPER(c))
PROPER(f(f(z0))) → c11(F(f(proper(z0))), PROPER(f(z0)))
ACTIVE(f(f(f(z0)))) → c2(F(f(mark(z0))), ACTIVE(f(f(z0))))
ACTIVE(f(f(f(z0)))) → c2(F(f(f(active(z0)))), ACTIVE(f(f(z0))))
S tuples:

TOP(found(z0)) → c5(TOP(active(z0)), ACTIVE(z0))
MATCH(X, z0) → c9(PROPER(z0))
F(ok(z0)) → c12(F(z0))
F(found(z0)) → c13(F(z0))
F(mark(z0)) → c14(F(z0))
ACTIVE(f(f(z0))) → c2(F(mark(z0)), ACTIVE(f(z0)))
TOP(mark(f(z0))) → c4(TOP(f(check(z0))), CHECK(f(z0)))
TOP(mark(z0)) → c4(TOP(start(match(f(X), z0))), CHECK(z0))
CHECK(f(f(z0))) → c6(F(f(check(z0))), CHECK(f(z0)))
CHECK(f(z0)) → c6(F(start(match(f(X), z0))), CHECK(z0))
CHECK(f(z1)) → c7(START(f(match(X, z1))), MATCH(f(X), f(z1)), F(X))
MATCH(f(f(z0)), f(f(z1))) → c8(F(f(match(z0, z1))), MATCH(f(z0), f(z1)))
MATCH(f(X), f(z0)) → c8(F(proper(z0)), MATCH(X, z0))
PROPER(f(c)) → c11(F(ok(c)), PROPER(c))
PROPER(f(f(z0))) → c11(F(f(proper(z0))), PROPER(f(z0)))
ACTIVE(f(f(f(z0)))) → c2(F(f(mark(z0))), ACTIVE(f(f(z0))))
ACTIVE(f(f(f(z0)))) → c2(F(f(f(active(z0)))), ACTIVE(f(f(z0))))
K tuples:

TOP(active(c)) → c3(TOP(mark(c)))
Defined Rule Symbols:

active, top, check, match, proper, f, start

Defined Pair Symbols:

TOP, MATCH, F, ACTIVE, CHECK, PROPER

Compound Symbols:

c3, c5, c9, c12, c13, c14, c2, c4, c6, c7, c8, c11

(33) 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 3.

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, 6, 7]
transitions:
mark0(0) → 0
c0() → 0
X0() → 0
ok0(0) → 0
found0(0) → 0
active0(0) → 1
top0(0) → 2
check0(0) → 3
match0(0, 0) → 4
proper0(0) → 5
f0(0) → 6
start0(0) → 7
check1(0) → 8
top1(8) → 2
X1() → 11
f1(11) → 10
match1(10, 0) → 9
start1(9) → 3
proper1(0) → 4
c1() → 12
ok1(12) → 5
f1(0) → 13
ok1(13) → 6
found1(0) → 7
f1(0) → 14
found1(14) → 6
active1(0) → 15
top1(15) → 2
f1(0) → 16
mark1(16) → 6
c1() → 18
mark1(18) → 17
top1(17) → 2
X2() → 21
f2(21) → 20
match2(20, 0) → 19
start2(19) → 8
ok1(12) → 4
ok1(13) → 13
ok1(13) → 14
ok1(13) → 16
found1(14) → 13
found1(14) → 14
found1(14) → 16
mark1(16) → 13
mark1(16) → 14
mark1(16) → 16
check2(18) → 22
top2(22) → 2
X3() → 25
f3(25) → 24
match3(24, 18) → 23
start3(23) → 22

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