(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
*(x, 0) → 0
*(x, s(y)) → +(*(x, y), x)
if(true, x, y) → x
if(false, x, y) → y
odd(0) → false
odd(s(0)) → true
odd(s(s(x))) → odd(x)
half(0) → 0
half(s(0)) → 0
half(s(s(x))) → s(half(x))
if(true, x, y) → true
if(false, x, y) → false
pow(x, y) → f(x, y, s(0))
f(x, 0, z) → z
f(x, s(y), z) → if(odd(s(y)), f(x, y, *(x, z)), f(*(x, x), half(s(y)), z))
Q is empty.
(1) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(2) Obligation:
Q DP problem:
The TRS P consists of the following rules:
-1(s(x), s(y)) → -1(x, y)
*1(x, s(y)) → *1(x, y)
ODD(s(s(x))) → ODD(x)
HALF(s(s(x))) → HALF(x)
POW(x, y) → F(x, y, s(0))
F(x, s(y), z) → IF(odd(s(y)), f(x, y, *(x, z)), f(*(x, x), half(s(y)), z))
F(x, s(y), z) → ODD(s(y))
F(x, s(y), z) → F(x, y, *(x, z))
F(x, s(y), z) → *1(x, z)
F(x, s(y), z) → F(*(x, x), half(s(y)), z)
F(x, s(y), z) → *1(x, x)
F(x, s(y), z) → HALF(s(y))
The TRS R consists of the following rules:
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
*(x, 0) → 0
*(x, s(y)) → +(*(x, y), x)
if(true, x, y) → x
if(false, x, y) → y
odd(0) → false
odd(s(0)) → true
odd(s(s(x))) → odd(x)
half(0) → 0
half(s(0)) → 0
half(s(s(x))) → s(half(x))
if(true, x, y) → true
if(false, x, y) → false
pow(x, y) → f(x, y, s(0))
f(x, 0, z) → z
f(x, s(y), z) → if(odd(s(y)), f(x, y, *(x, z)), f(*(x, x), half(s(y)), z))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 5 SCCs with 6 less nodes.
(4) Complex Obligation (AND)
(5) Obligation:
Q DP problem:
The TRS P consists of the following rules:
HALF(s(s(x))) → HALF(x)
The TRS R consists of the following rules:
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
*(x, 0) → 0
*(x, s(y)) → +(*(x, y), x)
if(true, x, y) → x
if(false, x, y) → y
odd(0) → false
odd(s(0)) → true
odd(s(s(x))) → odd(x)
half(0) → 0
half(s(0)) → 0
half(s(s(x))) → s(half(x))
if(true, x, y) → true
if(false, x, y) → false
pow(x, y) → f(x, y, s(0))
f(x, 0, z) → z
f(x, s(y), z) → if(odd(s(y)), f(x, y, *(x, z)), f(*(x, x), half(s(y)), z))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
ODD(s(s(x))) → ODD(x)
The TRS R consists of the following rules:
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
*(x, 0) → 0
*(x, s(y)) → +(*(x, y), x)
if(true, x, y) → x
if(false, x, y) → y
odd(0) → false
odd(s(0)) → true
odd(s(s(x))) → odd(x)
half(0) → 0
half(s(0)) → 0
half(s(s(x))) → s(half(x))
if(true, x, y) → true
if(false, x, y) → false
pow(x, y) → f(x, y, s(0))
f(x, 0, z) → z
f(x, s(y), z) → if(odd(s(y)), f(x, y, *(x, z)), f(*(x, x), half(s(y)), z))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(7) Obligation:
Q DP problem:
The TRS P consists of the following rules:
*1(x, s(y)) → *1(x, y)
The TRS R consists of the following rules:
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
*(x, 0) → 0
*(x, s(y)) → +(*(x, y), x)
if(true, x, y) → x
if(false, x, y) → y
odd(0) → false
odd(s(0)) → true
odd(s(s(x))) → odd(x)
half(0) → 0
half(s(0)) → 0
half(s(s(x))) → s(half(x))
if(true, x, y) → true
if(false, x, y) → false
pow(x, y) → f(x, y, s(0))
f(x, 0, z) → z
f(x, s(y), z) → if(odd(s(y)), f(x, y, *(x, z)), f(*(x, x), half(s(y)), z))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(x, s(y), z) → F(*(x, x), half(s(y)), z)
F(x, s(y), z) → F(x, y, *(x, z))
The TRS R consists of the following rules:
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
*(x, 0) → 0
*(x, s(y)) → +(*(x, y), x)
if(true, x, y) → x
if(false, x, y) → y
odd(0) → false
odd(s(0)) → true
odd(s(s(x))) → odd(x)
half(0) → 0
half(s(0)) → 0
half(s(s(x))) → s(half(x))
if(true, x, y) → true
if(false, x, y) → false
pow(x, y) → f(x, y, s(0))
f(x, 0, z) → z
f(x, s(y), z) → if(odd(s(y)), f(x, y, *(x, z)), f(*(x, x), half(s(y)), z))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(9) Obligation:
Q DP problem:
The TRS P consists of the following rules:
-1(s(x), s(y)) → -1(x, y)
The TRS R consists of the following rules:
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
*(x, 0) → 0
*(x, s(y)) → +(*(x, y), x)
if(true, x, y) → x
if(false, x, y) → y
odd(0) → false
odd(s(0)) → true
odd(s(s(x))) → odd(x)
half(0) → 0
half(s(0)) → 0
half(s(s(x))) → s(half(x))
if(true, x, y) → true
if(false, x, y) → false
pow(x, y) → f(x, y, s(0))
f(x, 0, z) → z
f(x, s(y), z) → if(odd(s(y)), f(x, y, *(x, z)), f(*(x, x), half(s(y)), z))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.