(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
a(lambda(x), y) → lambda(a(x, p(1, a(y, t))))
a(p(x, y), z) → p(a(x, z), a(y, z))
a(a(x, y), z) → a(x, a(y, z))
lambda(x) → x
a(x, y) → x
a(x, y) → y
p(x, y) → x
p(x, y) → y
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(lambda(z0), z1) → lambda(a(z0, p(1, a(z1, t))))
a(p(z0, z1), z2) → p(a(z0, z2), a(z1, z2))
a(a(z0, z1), z2) → a(z0, a(z1, z2))
a(z0, z1) → z0
a(z0, z1) → z1
lambda(z0) → z0
p(z0, z1) → z0
p(z0, z1) → z1
Tuples:
A(lambda(z0), z1) → c(LAMBDA(a(z0, p(1, a(z1, t)))), A(z0, p(1, a(z1, t))), P(1, a(z1, t)), A(z1, t))
A(p(z0, z1), z2) → c1(P(a(z0, z2), a(z1, z2)), A(z0, z2), A(z1, z2))
A(a(z0, z1), z2) → c2(A(z0, a(z1, z2)), A(z1, z2))
S tuples:
A(lambda(z0), z1) → c(LAMBDA(a(z0, p(1, a(z1, t)))), A(z0, p(1, a(z1, t))), P(1, a(z1, t)), A(z1, t))
A(p(z0, z1), z2) → c1(P(a(z0, z2), a(z1, z2)), A(z0, z2), A(z1, z2))
A(a(z0, z1), z2) → c2(A(z0, a(z1, z2)), A(z1, z2))
K tuples:none
Defined Rule Symbols:
a, lambda, p
Defined Pair Symbols:
A
Compound Symbols:
c, c1, c2
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
A(lambda(z0), z1) → c(LAMBDA(a(z0, p(1, a(z1, t)))), A(z0, p(1, a(z1, t))), P(1, a(z1, t)), A(z1, t))
A(p(z0, z1), z2) → c1(P(a(z0, z2), a(z1, z2)), A(z0, z2), A(z1, z2))
A(a(z0, z1), z2) → c2(A(z0, a(z1, z2)), A(z1, z2))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(lambda(z0), z1) → lambda(a(z0, p(1, a(z1, t))))
a(p(z0, z1), z2) → p(a(z0, z2), a(z1, z2))
a(a(z0, z1), z2) → a(z0, a(z1, z2))
a(z0, z1) → z0
a(z0, z1) → z1
lambda(z0) → z0
p(z0, z1) → z0
p(z0, z1) → z1
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
a, lambda, p
Defined Pair Symbols:none
Compound Symbols:none
(5) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(6) BOUNDS(O(1), O(1))