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

(0) Obligation:

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

b(y, z) → f(c(c(y, z, z), a, a))
b(b(z, y), a) → z
c(f(z), f(c(a, x, a)), y) → c(f(b(x, z)), c(z, y, a), a)

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

b(z0, z1) → f(c(c(z0, z1, z1), a, a))
b(b(z0, z1), a) → z0
c(f(z0), f(c(a, z1, a)), z2) → c(f(b(z1, z0)), c(z0, z2, a), a)

Tuples:

B(z0, z1) → c1(C(c(z0, z1, z1), a, a), C(z0, z1, z1))
C(f(z0), f(c(a, z1, a)), z2) → c3(C(f(b(z1, z0)), c(z0, z2, a), a), B(z1, z0), C(z0, z2, a))

S tuples:

B(z0, z1) → c1(C(c(z0, z1, z1), a, a), C(z0, z1, z1))
C(f(z0), f(c(a, z1, a)), z2) → c3(C(f(b(z1, z0)), c(z0, z2, a), a), B(z1, z0), C(z0, z2, a))

K tuples:none
Defined Rule Symbols:

b, c

Defined Pair Symbols:

B, C

Compound Symbols:

c1, c3

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

Removed 1 trailing tuple parts

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

b(z0, z1) → f(c(c(z0, z1, z1), a, a))
b(b(z0, z1), a) → z0
c(f(z0), f(c(a, z1, a)), z2) → c(f(b(z1, z0)), c(z0, z2, a), a)

Tuples:

C(f(z0), f(c(a, z1, a)), z2) → c3(C(f(b(z1, z0)), c(z0, z2, a), a), B(z1, z0), C(z0, z2, a))
B(z0, z1) → c1(C(z0, z1, z1))

S tuples:

C(f(z0), f(c(a, z1, a)), z2) → c3(C(f(b(z1, z0)), c(z0, z2, a), a), B(z1, z0), C(z0, z2, a))
B(z0, z1) → c1(C(z0, z1, z1))

K tuples:none
Defined Rule Symbols:

b, c

Defined Pair Symbols:

C, B

Compound Symbols:

c3, c1

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

Use narrowing to replace C(f(z0), f(c(a, z1, a)), z2) → c3(C(f(b(z1, z0)), c(z0, z2, a), a), B(z1, z0), C(z0, z2, a)) by

C(f(f(z0)), f(c(a, x1, a)), f(c(a, z1, a))) → c3(C(f(b(x1, f(z0))), c(f(b(z1, z0)), c(z0, a, a), a), a), B(x1, f(z0)), C(f(z0), f(c(a, z1, a)), a))
C(f(z1), f(c(a, z0, a)), x2) → c3(C(f(f(c(c(z0, z1, z1), a, a))), c(z1, x2, a), a), B(z0, z1), C(z1, x2, a))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

b(z0, z1) → f(c(c(z0, z1, z1), a, a))
b(b(z0, z1), a) → z0
c(f(z0), f(c(a, z1, a)), z2) → c(f(b(z1, z0)), c(z0, z2, a), a)

Tuples:

B(z0, z1) → c1(C(z0, z1, z1))
C(f(f(z0)), f(c(a, x1, a)), f(c(a, z1, a))) → c3(C(f(b(x1, f(z0))), c(f(b(z1, z0)), c(z0, a, a), a), a), B(x1, f(z0)), C(f(z0), f(c(a, z1, a)), a))
C(f(z1), f(c(a, z0, a)), x2) → c3(C(f(f(c(c(z0, z1, z1), a, a))), c(z1, x2, a), a), B(z0, z1), C(z1, x2, a))

S tuples:

B(z0, z1) → c1(C(z0, z1, z1))
C(f(f(z0)), f(c(a, x1, a)), f(c(a, z1, a))) → c3(C(f(b(x1, f(z0))), c(f(b(z1, z0)), c(z0, a, a), a), a), B(x1, f(z0)), C(f(z0), f(c(a, z1, a)), a))
C(f(z1), f(c(a, z0, a)), x2) → c3(C(f(f(c(c(z0, z1, z1), a, a))), c(z1, x2, a), a), B(z0, z1), C(z1, x2, a))

K tuples:none
Defined Rule Symbols:

b, c

Defined Pair Symbols:

B, C

Compound Symbols:

c1, c3

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

Removed 1 trailing tuple parts

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

b(z0, z1) → f(c(c(z0, z1, z1), a, a))
b(b(z0, z1), a) → z0
c(f(z0), f(c(a, z1, a)), z2) → c(f(b(z1, z0)), c(z0, z2, a), a)

Tuples:

B(z0, z1) → c1(C(z0, z1, z1))
C(f(z1), f(c(a, z0, a)), x2) → c3(C(f(f(c(c(z0, z1, z1), a, a))), c(z1, x2, a), a), B(z0, z1), C(z1, x2, a))
C(f(f(z0)), f(c(a, x1, a)), f(c(a, z1, a))) → c3(B(x1, f(z0)), C(f(z0), f(c(a, z1, a)), a))

S tuples:

B(z0, z1) → c1(C(z0, z1, z1))
C(f(z1), f(c(a, z0, a)), x2) → c3(C(f(f(c(c(z0, z1, z1), a, a))), c(z1, x2, a), a), B(z0, z1), C(z1, x2, a))
C(f(f(z0)), f(c(a, x1, a)), f(c(a, z1, a))) → c3(B(x1, f(z0)), C(f(z0), f(c(a, z1, a)), a))

K tuples:none
Defined Rule Symbols:

b, c

Defined Pair Symbols:

B, C

Compound Symbols:

c1, c3, c3

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

Use narrowing to replace C(f(z1), f(c(a, z0, a)), x2) → c3(C(f(f(c(c(z0, z1, z1), a, a))), c(z1, x2, a), a), B(z0, z1), C(z1, x2, a)) by

C(f(f(z0)), f(c(a, x1, a)), f(c(a, z1, a))) → c3(C(f(f(c(c(x1, f(z0), f(z0)), a, a))), c(f(b(z1, z0)), c(z0, a, a), a), a), B(x1, f(z0)), C(f(z0), f(c(a, z1, a)), a))
C(f(f(c(a, z1, a))), f(c(a, f(z0), a)), x2) → c3(C(f(f(c(c(f(b(z1, z0)), c(z0, f(c(a, z1, a)), a), a), a, a))), c(f(c(a, z1, a)), x2, a), a), B(f(z0), f(c(a, z1, a))), C(f(c(a, z1, a)), x2, a))
C(f(x0), f(c(a, x1, a)), x2) → c3(B(x1, x0))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

b(z0, z1) → f(c(c(z0, z1, z1), a, a))
b(b(z0, z1), a) → z0
c(f(z0), f(c(a, z1, a)), z2) → c(f(b(z1, z0)), c(z0, z2, a), a)

Tuples:

B(z0, z1) → c1(C(z0, z1, z1))
C(f(f(z0)), f(c(a, x1, a)), f(c(a, z1, a))) → c3(B(x1, f(z0)), C(f(z0), f(c(a, z1, a)), a))
C(f(f(z0)), f(c(a, x1, a)), f(c(a, z1, a))) → c3(C(f(f(c(c(x1, f(z0), f(z0)), a, a))), c(f(b(z1, z0)), c(z0, a, a), a), a), B(x1, f(z0)), C(f(z0), f(c(a, z1, a)), a))
C(f(f(c(a, z1, a))), f(c(a, f(z0), a)), x2) → c3(C(f(f(c(c(f(b(z1, z0)), c(z0, f(c(a, z1, a)), a), a), a, a))), c(f(c(a, z1, a)), x2, a), a), B(f(z0), f(c(a, z1, a))), C(f(c(a, z1, a)), x2, a))
C(f(x0), f(c(a, x1, a)), x2) → c3(B(x1, x0))

S tuples:

B(z0, z1) → c1(C(z0, z1, z1))
C(f(f(z0)), f(c(a, x1, a)), f(c(a, z1, a))) → c3(B(x1, f(z0)), C(f(z0), f(c(a, z1, a)), a))
C(f(f(z0)), f(c(a, x1, a)), f(c(a, z1, a))) → c3(C(f(f(c(c(x1, f(z0), f(z0)), a, a))), c(f(b(z1, z0)), c(z0, a, a), a), a), B(x1, f(z0)), C(f(z0), f(c(a, z1, a)), a))
C(f(f(c(a, z1, a))), f(c(a, f(z0), a)), x2) → c3(C(f(f(c(c(f(b(z1, z0)), c(z0, f(c(a, z1, a)), a), a), a, a))), c(f(c(a, z1, a)), x2, a), a), B(f(z0), f(c(a, z1, a))), C(f(c(a, z1, a)), x2, a))
C(f(x0), f(c(a, x1, a)), x2) → c3(B(x1, x0))

K tuples:none
Defined Rule Symbols:

b, c

Defined Pair Symbols:

B, C

Compound Symbols:

c1, c3, c3, c3

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

Removed 1 trailing tuple parts

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

b(z0, z1) → f(c(c(z0, z1, z1), a, a))
b(b(z0, z1), a) → z0
c(f(z0), f(c(a, z1, a)), z2) → c(f(b(z1, z0)), c(z0, z2, a), a)

Tuples:

B(z0, z1) → c1(C(z0, z1, z1))
C(f(f(z0)), f(c(a, x1, a)), f(c(a, z1, a))) → c3(B(x1, f(z0)), C(f(z0), f(c(a, z1, a)), a))
C(f(f(c(a, z1, a))), f(c(a, f(z0), a)), x2) → c3(C(f(f(c(c(f(b(z1, z0)), c(z0, f(c(a, z1, a)), a), a), a, a))), c(f(c(a, z1, a)), x2, a), a), B(f(z0), f(c(a, z1, a))), C(f(c(a, z1, a)), x2, a))
C(f(x0), f(c(a, x1, a)), x2) → c3(B(x1, x0))

S tuples:

B(z0, z1) → c1(C(z0, z1, z1))
C(f(f(z0)), f(c(a, x1, a)), f(c(a, z1, a))) → c3(B(x1, f(z0)), C(f(z0), f(c(a, z1, a)), a))
C(f(f(c(a, z1, a))), f(c(a, f(z0), a)), x2) → c3(C(f(f(c(c(f(b(z1, z0)), c(z0, f(c(a, z1, a)), a), a), a, a))), c(f(c(a, z1, a)), x2, a), a), B(f(z0), f(c(a, z1, a))), C(f(c(a, z1, a)), x2, a))
C(f(x0), f(c(a, x1, a)), x2) → c3(B(x1, x0))

K tuples:none
Defined Rule Symbols:

b, c

Defined Pair Symbols:

B, C

Compound Symbols:

c1, c3, c3, c3

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

Use instantiation to replace C(f(f(c(a, z1, a))), f(c(a, f(z0), a)), x2) → c3(C(f(f(c(c(f(b(z1, z0)), c(z0, f(c(a, z1, a)), a), a), a, a))), c(f(c(a, z1, a)), x2, a), a), B(f(z0), f(c(a, z1, a))), C(f(c(a, z1, a)), x2, a)) by

C(f(f(c(a, z0, a))), f(c(a, f(z1), a)), f(c(a, f(z1), a))) → c3(C(f(f(c(c(f(b(z0, z1)), c(z1, f(c(a, z0, a)), a), a), a, a))), c(f(c(a, z0, a)), f(c(a, f(z1), a)), a), a), B(f(z1), f(c(a, z0, a))), C(f(c(a, z0, a)), f(c(a, f(z1), a)), a))
C(f(f(c(a, z0, a))), f(c(a, f(z1), a)), a) → c3(C(f(f(c(c(f(b(z0, z1)), c(z1, f(c(a, z0, a)), a), a), a, a))), c(f(c(a, z0, a)), a, a), a), B(f(z1), f(c(a, z0, a))), C(f(c(a, z0, a)), a, a))
C(f(f(c(a, a, a))), f(c(a, f(z1), a)), a) → c3(C(f(f(c(c(f(b(a, z1)), c(z1, f(c(a, a, a)), a), a), a, a))), c(f(c(a, a, a)), a, a), a), B(f(z1), f(c(a, a, a))), C(f(c(a, a, a)), a, a))

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

b(z0, z1) → f(c(c(z0, z1, z1), a, a))
b(b(z0, z1), a) → z0
c(f(z0), f(c(a, z1, a)), z2) → c(f(b(z1, z0)), c(z0, z2, a), a)

Tuples:

B(z0, z1) → c1(C(z0, z1, z1))
C(f(f(z0)), f(c(a, x1, a)), f(c(a, z1, a))) → c3(B(x1, f(z0)), C(f(z0), f(c(a, z1, a)), a))
C(f(x0), f(c(a, x1, a)), x2) → c3(B(x1, x0))
C(f(f(c(a, z0, a))), f(c(a, f(z1), a)), f(c(a, f(z1), a))) → c3(C(f(f(c(c(f(b(z0, z1)), c(z1, f(c(a, z0, a)), a), a), a, a))), c(f(c(a, z0, a)), f(c(a, f(z1), a)), a), a), B(f(z1), f(c(a, z0, a))), C(f(c(a, z0, a)), f(c(a, f(z1), a)), a))
C(f(f(c(a, z0, a))), f(c(a, f(z1), a)), a) → c3(C(f(f(c(c(f(b(z0, z1)), c(z1, f(c(a, z0, a)), a), a), a, a))), c(f(c(a, z0, a)), a, a), a), B(f(z1), f(c(a, z0, a))), C(f(c(a, z0, a)), a, a))
C(f(f(c(a, a, a))), f(c(a, f(z1), a)), a) → c3(C(f(f(c(c(f(b(a, z1)), c(z1, f(c(a, a, a)), a), a), a, a))), c(f(c(a, a, a)), a, a), a), B(f(z1), f(c(a, a, a))), C(f(c(a, a, a)), a, a))

S tuples:

B(z0, z1) → c1(C(z0, z1, z1))
C(f(f(z0)), f(c(a, x1, a)), f(c(a, z1, a))) → c3(B(x1, f(z0)), C(f(z0), f(c(a, z1, a)), a))
C(f(x0), f(c(a, x1, a)), x2) → c3(B(x1, x0))
C(f(f(c(a, z0, a))), f(c(a, f(z1), a)), f(c(a, f(z1), a))) → c3(C(f(f(c(c(f(b(z0, z1)), c(z1, f(c(a, z0, a)), a), a), a, a))), c(f(c(a, z0, a)), f(c(a, f(z1), a)), a), a), B(f(z1), f(c(a, z0, a))), C(f(c(a, z0, a)), f(c(a, f(z1), a)), a))
C(f(f(c(a, z0, a))), f(c(a, f(z1), a)), a) → c3(C(f(f(c(c(f(b(z0, z1)), c(z1, f(c(a, z0, a)), a), a), a, a))), c(f(c(a, z0, a)), a, a), a), B(f(z1), f(c(a, z0, a))), C(f(c(a, z0, a)), a, a))
C(f(f(c(a, a, a))), f(c(a, f(z1), a)), a) → c3(C(f(f(c(c(f(b(a, z1)), c(z1, f(c(a, a, a)), a), a), a, a))), c(f(c(a, a, a)), a, a), a), B(f(z1), f(c(a, a, a))), C(f(c(a, a, a)), a, a))

K tuples:none
Defined Rule Symbols:

b, c

Defined Pair Symbols:

B, C

Compound Symbols:

c1, c3, c3, c3

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

Removed 4 trailing tuple parts

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

b(z0, z1) → f(c(c(z0, z1, z1), a, a))
b(b(z0, z1), a) → z0
c(f(z0), f(c(a, z1, a)), z2) → c(f(b(z1, z0)), c(z0, z2, a), a)

Tuples:

B(z0, z1) → c1(C(z0, z1, z1))
C(f(f(z0)), f(c(a, x1, a)), f(c(a, z1, a))) → c3(B(x1, f(z0)), C(f(z0), f(c(a, z1, a)), a))
C(f(x0), f(c(a, x1, a)), x2) → c3(B(x1, x0))
C(f(f(c(a, z0, a))), f(c(a, f(z1), a)), f(c(a, f(z1), a))) → c3(C(f(f(c(c(f(b(z0, z1)), c(z1, f(c(a, z0, a)), a), a), a, a))), c(f(c(a, z0, a)), f(c(a, f(z1), a)), a), a), B(f(z1), f(c(a, z0, a))), C(f(c(a, z0, a)), f(c(a, f(z1), a)), a))
C(f(f(c(a, z0, a))), f(c(a, f(z1), a)), a) → c3(B(f(z1), f(c(a, z0, a))))
C(f(f(c(a, a, a))), f(c(a, f(z1), a)), a) → c3(B(f(z1), f(c(a, a, a))))

S tuples:

B(z0, z1) → c1(C(z0, z1, z1))
C(f(f(z0)), f(c(a, x1, a)), f(c(a, z1, a))) → c3(B(x1, f(z0)), C(f(z0), f(c(a, z1, a)), a))
C(f(x0), f(c(a, x1, a)), x2) → c3(B(x1, x0))
C(f(f(c(a, z0, a))), f(c(a, f(z1), a)), f(c(a, f(z1), a))) → c3(C(f(f(c(c(f(b(z0, z1)), c(z1, f(c(a, z0, a)), a), a), a, a))), c(f(c(a, z0, a)), f(c(a, f(z1), a)), a), a), B(f(z1), f(c(a, z0, a))), C(f(c(a, z0, a)), f(c(a, f(z1), a)), a))
C(f(f(c(a, z0, a))), f(c(a, f(z1), a)), a) → c3(B(f(z1), f(c(a, z0, a))))
C(f(f(c(a, a, a))), f(c(a, f(z1), a)), a) → c3(B(f(z1), f(c(a, a, a))))

K tuples:none
Defined Rule Symbols:

b, c

Defined Pair Symbols:

B, C

Compound Symbols:

c1, c3, c3, c3

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

Used rewriting to replace C(f(f(c(a, z0, a))), f(c(a, f(z1), a)), f(c(a, f(z1), a))) → c3(C(f(f(c(c(f(b(z0, z1)), c(z1, f(c(a, z0, a)), a), a), a, a))), c(f(c(a, z0, a)), f(c(a, f(z1), a)), a), a), B(f(z1), f(c(a, z0, a))), C(f(c(a, z0, a)), f(c(a, f(z1), a)), a)) by C(f(f(c(a, z0, a))), f(c(a, f(z1), a)), f(c(a, f(z1), a))) → c3(C(f(f(c(c(f(b(z0, z1)), c(z1, f(c(a, z0, a)), a), a), a, a))), c(f(b(f(z1), c(a, z0, a))), c(c(a, z0, a), a, a), a), a), B(f(z1), f(c(a, z0, a))), C(f(c(a, z0, a)), f(c(a, f(z1), a)), a))

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

b(z0, z1) → f(c(c(z0, z1, z1), a, a))
b(b(z0, z1), a) → z0
c(f(z0), f(c(a, z1, a)), z2) → c(f(b(z1, z0)), c(z0, z2, a), a)

Tuples:

B(z0, z1) → c1(C(z0, z1, z1))
C(f(f(z0)), f(c(a, x1, a)), f(c(a, z1, a))) → c3(B(x1, f(z0)), C(f(z0), f(c(a, z1, a)), a))
C(f(x0), f(c(a, x1, a)), x2) → c3(B(x1, x0))
C(f(f(c(a, z0, a))), f(c(a, f(z1), a)), a) → c3(B(f(z1), f(c(a, z0, a))))
C(f(f(c(a, a, a))), f(c(a, f(z1), a)), a) → c3(B(f(z1), f(c(a, a, a))))
C(f(f(c(a, z0, a))), f(c(a, f(z1), a)), f(c(a, f(z1), a))) → c3(C(f(f(c(c(f(b(z0, z1)), c(z1, f(c(a, z0, a)), a), a), a, a))), c(f(b(f(z1), c(a, z0, a))), c(c(a, z0, a), a, a), a), a), B(f(z1), f(c(a, z0, a))), C(f(c(a, z0, a)), f(c(a, f(z1), a)), a))

S tuples:

B(z0, z1) → c1(C(z0, z1, z1))
C(f(f(z0)), f(c(a, x1, a)), f(c(a, z1, a))) → c3(B(x1, f(z0)), C(f(z0), f(c(a, z1, a)), a))
C(f(x0), f(c(a, x1, a)), x2) → c3(B(x1, x0))
C(f(f(c(a, z0, a))), f(c(a, f(z1), a)), a) → c3(B(f(z1), f(c(a, z0, a))))
C(f(f(c(a, a, a))), f(c(a, f(z1), a)), a) → c3(B(f(z1), f(c(a, a, a))))
C(f(f(c(a, z0, a))), f(c(a, f(z1), a)), f(c(a, f(z1), a))) → c3(C(f(f(c(c(f(b(z0, z1)), c(z1, f(c(a, z0, a)), a), a), a, a))), c(f(b(f(z1), c(a, z0, a))), c(c(a, z0, a), a, a), a), a), B(f(z1), f(c(a, z0, a))), C(f(c(a, z0, a)), f(c(a, f(z1), a)), a))

K tuples:none
Defined Rule Symbols:

b, c

Defined Pair Symbols:

B, C

Compound Symbols:

c1, c3, c3, c3

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

Removed 1 trailing tuple parts

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

b(z0, z1) → f(c(c(z0, z1, z1), a, a))
b(b(z0, z1), a) → z0
c(f(z0), f(c(a, z1, a)), z2) → c(f(b(z1, z0)), c(z0, z2, a), a)

Tuples:

B(z0, z1) → c1(C(z0, z1, z1))
C(f(f(z0)), f(c(a, x1, a)), f(c(a, z1, a))) → c3(B(x1, f(z0)), C(f(z0), f(c(a, z1, a)), a))
C(f(x0), f(c(a, x1, a)), x2) → c3(B(x1, x0))
C(f(f(c(a, z0, a))), f(c(a, f(z1), a)), a) → c3(B(f(z1), f(c(a, z0, a))))
C(f(f(c(a, a, a))), f(c(a, f(z1), a)), a) → c3(B(f(z1), f(c(a, a, a))))
C(f(f(c(a, z0, a))), f(c(a, f(z1), a)), f(c(a, f(z1), a))) → c3(B(f(z1), f(c(a, z0, a))), C(f(c(a, z0, a)), f(c(a, f(z1), a)), a))

S tuples:

B(z0, z1) → c1(C(z0, z1, z1))
C(f(f(z0)), f(c(a, x1, a)), f(c(a, z1, a))) → c3(B(x1, f(z0)), C(f(z0), f(c(a, z1, a)), a))
C(f(x0), f(c(a, x1, a)), x2) → c3(B(x1, x0))
C(f(f(c(a, z0, a))), f(c(a, f(z1), a)), a) → c3(B(f(z1), f(c(a, z0, a))))
C(f(f(c(a, a, a))), f(c(a, f(z1), a)), a) → c3(B(f(z1), f(c(a, a, a))))
C(f(f(c(a, z0, a))), f(c(a, f(z1), a)), f(c(a, f(z1), a))) → c3(B(f(z1), f(c(a, z0, a))), C(f(c(a, z0, a)), f(c(a, f(z1), a)), a))

K tuples:none
Defined Rule Symbols:

b, c

Defined Pair Symbols:

B, C

Compound Symbols:

c1, c3, c3

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

Use instantiation to replace C(f(x0), f(c(a, x1, a)), x2) → c3(B(x1, x0)) by

C(f(z0), f(c(a, z1, a)), f(c(a, z1, a))) → c3(B(z1, z0))
C(f(x0), f(c(a, x2, a)), a) → c3(B(x2, x0))
C(f(c(a, x0, a)), f(c(a, f(x1), a)), a) → c3(B(f(x1), c(a, x0, a)))

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

b(z0, z1) → f(c(c(z0, z1, z1), a, a))
b(b(z0, z1), a) → z0
c(f(z0), f(c(a, z1, a)), z2) → c(f(b(z1, z0)), c(z0, z2, a), a)

Tuples:

B(z0, z1) → c1(C(z0, z1, z1))
C(f(f(z0)), f(c(a, x1, a)), f(c(a, z1, a))) → c3(B(x1, f(z0)), C(f(z0), f(c(a, z1, a)), a))
C(f(f(c(a, z0, a))), f(c(a, f(z1), a)), a) → c3(B(f(z1), f(c(a, z0, a))))
C(f(f(c(a, a, a))), f(c(a, f(z1), a)), a) → c3(B(f(z1), f(c(a, a, a))))
C(f(f(c(a, z0, a))), f(c(a, f(z1), a)), f(c(a, f(z1), a))) → c3(B(f(z1), f(c(a, z0, a))), C(f(c(a, z0, a)), f(c(a, f(z1), a)), a))
C(f(z0), f(c(a, z1, a)), f(c(a, z1, a))) → c3(B(z1, z0))
C(f(x0), f(c(a, x2, a)), a) → c3(B(x2, x0))
C(f(c(a, x0, a)), f(c(a, f(x1), a)), a) → c3(B(f(x1), c(a, x0, a)))

S tuples:

B(z0, z1) → c1(C(z0, z1, z1))
C(f(f(z0)), f(c(a, x1, a)), f(c(a, z1, a))) → c3(B(x1, f(z0)), C(f(z0), f(c(a, z1, a)), a))
C(f(f(c(a, z0, a))), f(c(a, f(z1), a)), a) → c3(B(f(z1), f(c(a, z0, a))))
C(f(f(c(a, a, a))), f(c(a, f(z1), a)), a) → c3(B(f(z1), f(c(a, a, a))))
C(f(f(c(a, z0, a))), f(c(a, f(z1), a)), f(c(a, f(z1), a))) → c3(B(f(z1), f(c(a, z0, a))), C(f(c(a, z0, a)), f(c(a, f(z1), a)), a))
C(f(z0), f(c(a, z1, a)), f(c(a, z1, a))) → c3(B(z1, z0))
C(f(x0), f(c(a, x2, a)), a) → c3(B(x2, x0))
C(f(c(a, x0, a)), f(c(a, f(x1), a)), a) → c3(B(f(x1), c(a, x0, a)))

K tuples:none
Defined Rule Symbols:

b, c

Defined Pair Symbols:

B, C

Compound Symbols:

c1, c3, c3

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

Use forward instantiation to replace B(z0, z1) → c1(C(z0, z1, z1)) by

B(f(f(y0)), f(c(a, y1, a))) → c1(C(f(f(y0)), f(c(a, y1, a)), f(c(a, y1, a))))
B(f(f(c(a, y0, a))), f(c(a, f(y1), a))) → c1(C(f(f(c(a, y0, a))), f(c(a, f(y1), a)), f(c(a, f(y1), a))))
B(f(y0), f(c(a, y1, a))) → c1(C(f(y0), f(c(a, y1, a)), f(c(a, y1, a))))

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

b(z0, z1) → f(c(c(z0, z1, z1), a, a))
b(b(z0, z1), a) → z0
c(f(z0), f(c(a, z1, a)), z2) → c(f(b(z1, z0)), c(z0, z2, a), a)

Tuples:

C(f(f(z0)), f(c(a, x1, a)), f(c(a, z1, a))) → c3(B(x1, f(z0)), C(f(z0), f(c(a, z1, a)), a))
C(f(f(c(a, z0, a))), f(c(a, f(z1), a)), a) → c3(B(f(z1), f(c(a, z0, a))))
C(f(f(c(a, a, a))), f(c(a, f(z1), a)), a) → c3(B(f(z1), f(c(a, a, a))))
C(f(f(c(a, z0, a))), f(c(a, f(z1), a)), f(c(a, f(z1), a))) → c3(B(f(z1), f(c(a, z0, a))), C(f(c(a, z0, a)), f(c(a, f(z1), a)), a))
C(f(z0), f(c(a, z1, a)), f(c(a, z1, a))) → c3(B(z1, z0))
C(f(x0), f(c(a, x2, a)), a) → c3(B(x2, x0))
C(f(c(a, x0, a)), f(c(a, f(x1), a)), a) → c3(B(f(x1), c(a, x0, a)))
B(f(f(y0)), f(c(a, y1, a))) → c1(C(f(f(y0)), f(c(a, y1, a)), f(c(a, y1, a))))
B(f(f(c(a, y0, a))), f(c(a, f(y1), a))) → c1(C(f(f(c(a, y0, a))), f(c(a, f(y1), a)), f(c(a, f(y1), a))))
B(f(y0), f(c(a, y1, a))) → c1(C(f(y0), f(c(a, y1, a)), f(c(a, y1, a))))

S tuples:

C(f(f(z0)), f(c(a, x1, a)), f(c(a, z1, a))) → c3(B(x1, f(z0)), C(f(z0), f(c(a, z1, a)), a))
C(f(f(c(a, z0, a))), f(c(a, f(z1), a)), a) → c3(B(f(z1), f(c(a, z0, a))))
C(f(f(c(a, a, a))), f(c(a, f(z1), a)), a) → c3(B(f(z1), f(c(a, a, a))))
C(f(f(c(a, z0, a))), f(c(a, f(z1), a)), f(c(a, f(z1), a))) → c3(B(f(z1), f(c(a, z0, a))), C(f(c(a, z0, a)), f(c(a, f(z1), a)), a))
C(f(z0), f(c(a, z1, a)), f(c(a, z1, a))) → c3(B(z1, z0))
C(f(x0), f(c(a, x2, a)), a) → c3(B(x2, x0))
C(f(c(a, x0, a)), f(c(a, f(x1), a)), a) → c3(B(f(x1), c(a, x0, a)))
B(f(f(y0)), f(c(a, y1, a))) → c1(C(f(f(y0)), f(c(a, y1, a)), f(c(a, y1, a))))
B(f(f(c(a, y0, a))), f(c(a, f(y1), a))) → c1(C(f(f(c(a, y0, a))), f(c(a, f(y1), a)), f(c(a, f(y1), a))))
B(f(y0), f(c(a, y1, a))) → c1(C(f(y0), f(c(a, y1, a)), f(c(a, y1, a))))

K tuples:none
Defined Rule Symbols:

b, c

Defined Pair Symbols:

C, B

Compound Symbols:

c3, c3, c1

(25) CdtUnreachableProof (EQUIVALENT transformation)

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

C(f(f(z0)), f(c(a, x1, a)), f(c(a, z1, a))) → c3(B(x1, f(z0)), C(f(z0), f(c(a, z1, a)), a))
C(f(f(c(a, z0, a))), f(c(a, f(z1), a)), a) → c3(B(f(z1), f(c(a, z0, a))))
C(f(f(c(a, a, a))), f(c(a, f(z1), a)), a) → c3(B(f(z1), f(c(a, a, a))))
C(f(f(c(a, z0, a))), f(c(a, f(z1), a)), f(c(a, f(z1), a))) → c3(B(f(z1), f(c(a, z0, a))), C(f(c(a, z0, a)), f(c(a, f(z1), a)), a))
C(f(z0), f(c(a, z1, a)), f(c(a, z1, a))) → c3(B(z1, z0))
C(f(x0), f(c(a, x2, a)), a) → c3(B(x2, x0))
C(f(c(a, x0, a)), f(c(a, f(x1), a)), a) → c3(B(f(x1), c(a, x0, a)))
B(f(f(y0)), f(c(a, y1, a))) → c1(C(f(f(y0)), f(c(a, y1, a)), f(c(a, y1, a))))
B(f(f(c(a, y0, a))), f(c(a, f(y1), a))) → c1(C(f(f(c(a, y0, a))), f(c(a, f(y1), a)), f(c(a, f(y1), a))))
B(f(y0), f(c(a, y1, a))) → c1(C(f(y0), f(c(a, y1, a)), f(c(a, y1, a))))

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

b(z0, z1) → f(c(c(z0, z1, z1), a, a))
b(b(z0, z1), a) → z0
c(f(z0), f(c(a, z1, a)), z2) → c(f(b(z1, z0)), c(z0, z2, a), a)

Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

b, c

Defined Pair Symbols:none

Compound Symbols:none

(27) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(28) BOUNDS(O(1), O(1))