(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
fact(X) → if(zero(X), s(0), 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) → X
if(false, X, Y) → Y
zero(0) → true
zero(s(X)) → false
p(s(X)) → X
Q is empty.
(1) Overlay + Local Confluence (EQUIVALENT transformation)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
fact(X) → if(zero(X), s(0), 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) → X
if(false, X, Y) → Y
zero(0) → true
zero(s(X)) → false
p(s(X)) → X
The set Q consists of the following terms:
fact(x0)
add(0, x0)
add(s(x0), x1)
prod(0, x0)
prod(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
zero(0)
zero(s(x0))
p(s(x0))
(3) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
FACT(X) → IF(zero(X), s(0), prod(X, fact(p(X))))
FACT(X) → ZERO(X)
FACT(X) → PROD(X, fact(p(X)))
FACT(X) → FACT(p(X))
FACT(X) → P(X)
ADD(s(X), Y) → ADD(X, Y)
PROD(s(X), Y) → ADD(Y, prod(X, Y))
PROD(s(X), Y) → PROD(X, Y)
The TRS R consists of the following rules:
fact(X) → if(zero(X), s(0), 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) → X
if(false, X, Y) → Y
zero(0) → true
zero(s(X)) → false
p(s(X)) → X
The set Q consists of the following terms:
fact(x0)
add(0, x0)
add(s(x0), x1)
prod(0, x0)
prod(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
zero(0)
zero(s(x0))
p(s(x0))
We have to consider all minimal (P,Q,R)-chains.
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 5 less nodes.
(6) Complex Obligation (AND)
(7) Obligation:
Q DP problem:
The TRS P consists of the following rules:
ADD(s(X), Y) → ADD(X, Y)
The TRS R consists of the following rules:
fact(X) → if(zero(X), s(0), 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) → X
if(false, X, Y) → Y
zero(0) → true
zero(s(X)) → false
p(s(X)) → X
The set Q consists of the following terms:
fact(x0)
add(0, x0)
add(s(x0), x1)
prod(0, x0)
prod(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
zero(0)
zero(s(x0))
p(s(x0))
We have to consider all minimal (P,Q,R)-chains.
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
PROD(s(X), Y) → PROD(X, Y)
The TRS R consists of the following rules:
fact(X) → if(zero(X), s(0), 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) → X
if(false, X, Y) → Y
zero(0) → true
zero(s(X)) → false
p(s(X)) → X
The set Q consists of the following terms:
fact(x0)
add(0, x0)
add(s(x0), x1)
prod(0, x0)
prod(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
zero(0)
zero(s(x0))
p(s(x0))
We have to consider all minimal (P,Q,R)-chains.
(9) Obligation:
Q DP problem:
The TRS P consists of the following rules:
FACT(X) → FACT(p(X))
The TRS R consists of the following rules:
fact(X) → if(zero(X), s(0), 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) → X
if(false, X, Y) → Y
zero(0) → true
zero(s(X)) → false
p(s(X)) → X
The set Q consists of the following terms:
fact(x0)
add(0, x0)
add(s(x0), x1)
prod(0, x0)
prod(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
zero(0)
zero(s(x0))
p(s(x0))
We have to consider all minimal (P,Q,R)-chains.