(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(c(c(a, y, a), b(x, z), a)) → b(y, f(c(f(a), z, z)))
f(b(b(x, f(y)), z)) → c(z, x, f(b(b(f(a), y), y)))
c(b(a, a), b(y, z), x) → b(a, b(z, z))
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(c(c(a, z0, a), b(z1, z2), a)) → b(z0, f(c(f(a), z2, z2)))
f(b(b(z0, f(z1)), z2)) → c(z2, z0, f(b(b(f(a), z1), z1)))
c(b(a, a), b(z0, z1), z2) → b(a, b(z1, z1))
Tuples:
F(c(c(a, z0, a), b(z1, z2), a)) → c1(F(c(f(a), z2, z2)), C(f(a), z2, z2), F(a))
F(b(b(z0, f(z1)), z2)) → c2(C(z2, z0, f(b(b(f(a), z1), z1))), F(b(b(f(a), z1), z1)), F(a))
C(b(a, a), b(z0, z1), z2) → c3
S tuples:
F(c(c(a, z0, a), b(z1, z2), a)) → c1(F(c(f(a), z2, z2)), C(f(a), z2, z2), F(a))
F(b(b(z0, f(z1)), z2)) → c2(C(z2, z0, f(b(b(f(a), z1), z1))), F(b(b(f(a), z1), z1)), F(a))
C(b(a, a), b(z0, z1), z2) → c3
K tuples:none
Defined Rule Symbols:
f, c
Defined Pair Symbols:
F, C
Compound Symbols:
c1, c2, c3
(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing nodes:
F(c(c(a, z0, a), b(z1, z2), a)) → c1(F(c(f(a), z2, z2)), C(f(a), z2, z2), F(a))
C(b(a, a), b(z0, z1), z2) → c3
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(c(c(a, z0, a), b(z1, z2), a)) → b(z0, f(c(f(a), z2, z2)))
f(b(b(z0, f(z1)), z2)) → c(z2, z0, f(b(b(f(a), z1), z1)))
c(b(a, a), b(z0, z1), z2) → b(a, b(z1, z1))
Tuples:
F(b(b(z0, f(z1)), z2)) → c2(C(z2, z0, f(b(b(f(a), z1), z1))), F(b(b(f(a), z1), z1)), F(a))
S tuples:
F(b(b(z0, f(z1)), z2)) → c2(C(z2, z0, f(b(b(f(a), z1), z1))), F(b(b(f(a), z1), z1)), F(a))
K tuples:none
Defined Rule Symbols:
f, c
Defined Pair Symbols:
F
Compound Symbols:
c2
(5) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
F(b(b(z0, f(z1)), z2)) → c2(C(z2, z0, f(b(b(f(a), z1), z1))), F(b(b(f(a), z1), z1)), F(a))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(c(c(a, z0, a), b(z1, z2), a)) → b(z0, f(c(f(a), z2, z2)))
f(b(b(z0, f(z1)), z2)) → c(z2, z0, f(b(b(f(a), z1), z1)))
c(b(a, a), b(z0, z1), z2) → b(a, b(z1, z1))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
f, c
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(8) BOUNDS(1, 1)