(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
fact(X) → if(zero(X), n__s(n__0), n__prod(X, n__fact(n__p(X))))
add(0, X) → X
add(s(X), Y) → s(add(X, Y))
prod(0, X) → 0
prod(s(X), Y) → add(Y, prod(X, Y))
if(true, X, Y) → activate(X)
if(false, X, Y) → activate(Y)
zero(0) → true
zero(s(X)) → false
p(s(X)) → X
s(X) → n__s(X)
0 → n__0
prod(X1, X2) → n__prod(X1, X2)
fact(X) → n__fact(X)
p(X) → n__p(X)
activate(n__s(X)) → s(activate(X))
activate(n__0) → 0
activate(n__prod(X1, X2)) → prod(activate(X1), activate(X2))
activate(n__fact(X)) → fact(activate(X))
activate(n__p(X)) → p(activate(X))
activate(X) → X
Rewrite Strategy: FULL
(1) DecreasingLoopProof (EQUIVALENT transformation)
The following loop(s) give(s) rise to the lower bound Ω(2n):
The rewrite sequence
activate(n__s(n__fact(X31510_3))) →+ s(if(zero(activate(X31510_3)), n__s(n__0), n__prod(activate(X31510_3), n__fact(n__p(activate(X31510_3))))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0].
The pumping substitution is [X31510_3 / n__s(n__fact(X31510_3))].
The result substitution is [ ].
The rewrite sequence
activate(n__s(n__fact(X31510_3))) →+ s(if(zero(activate(X31510_3)), n__s(n__0), n__prod(activate(X31510_3), n__fact(n__p(activate(X31510_3))))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,2,0].
The pumping substitution is [X31510_3 / n__s(n__fact(X31510_3))].
The result substitution is [ ].
(2) BOUNDS(2^n, INF)