(0) Obligation:

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

fact(X) → if(zero(X), n__s(0), n__prod(X, fact(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)
prod(X1, X2) → n__prod(X1, X2)
activate(n__s(X)) → s(X)
activate(n__prod(X1, X2)) → prod(X1, X2)
activate(X) → X

Rewrite Strategy: FULL

(1) InfiniteLowerBoundProof (EQUIVALENT transformation)

The loop following loop proves infinite runtime complexity:
The rewrite sequence
fact(X) →+ if(zero(X), n__s(0), n__prod(X, fact(p(X))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [2,1].
The pumping substitution is [ ].
The result substitution is [X / p(X)].