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