(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

-(x, 0) → x
-(s(x), s(y)) → -(x, y)
min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
twice(0) → 0
twice(s(x)) → s(s(twice(x)))
f(s(x), s(y)) → f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) → f(-(x, min(x, y)), s(twice(min(x, y))))

Q is empty.

(1) AAECC Innermost (EQUIVALENT transformation)

We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is

-(x, 0) → x
-(s(x), s(y)) → -(x, y)
min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
twice(0) → 0
twice(s(x)) → s(s(twice(x)))

The TRS R 2 is

f(s(x), s(y)) → f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) → f(-(x, min(x, y)), s(twice(min(x, y))))

The signature Sigma is {f}

(2) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

-(x, 0) → x
-(s(x), s(y)) → -(x, y)
min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
twice(0) → 0
twice(s(x)) → s(s(twice(x)))
f(s(x), s(y)) → f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) → f(-(x, min(x, y)), s(twice(min(x, y))))

The set Q consists of the following terms:

-(x0, 0)
-(s(x0), s(x1))
min(x0, 0)
min(0, x0)
min(s(x0), s(x1))
twice(0)
twice(s(x0))
f(s(x0), s(x1))

(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:

-1(s(x), s(y)) → -1(x, y)
MIN(s(x), s(y)) → MIN(x, y)
TWICE(s(x)) → TWICE(x)
F(s(x), s(y)) → F(-(y, min(x, y)), s(twice(min(x, y))))
F(s(x), s(y)) → -1(y, min(x, y))
F(s(x), s(y)) → MIN(x, y)
F(s(x), s(y)) → TWICE(min(x, y))
F(s(x), s(y)) → F(-(x, min(x, y)), s(twice(min(x, y))))
F(s(x), s(y)) → -1(x, min(x, y))

The TRS R consists of the following rules:

-(x, 0) → x
-(s(x), s(y)) → -(x, y)
min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
twice(0) → 0
twice(s(x)) → s(s(twice(x)))
f(s(x), s(y)) → f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) → f(-(x, min(x, y)), s(twice(min(x, y))))

The set Q consists of the following terms:

-(x0, 0)
-(s(x0), s(x1))
min(x0, 0)
min(0, x0)
min(s(x0), s(x1))
twice(0)
twice(s(x0))
f(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 4 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

Q DP problem:
The TRS P consists of the following rules:

TWICE(s(x)) → TWICE(x)

The TRS R consists of the following rules:

-(x, 0) → x
-(s(x), s(y)) → -(x, y)
min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
twice(0) → 0
twice(s(x)) → s(s(twice(x)))
f(s(x), s(y)) → f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) → f(-(x, min(x, y)), s(twice(min(x, y))))

The set Q consists of the following terms:

-(x0, 0)
-(s(x0), s(x1))
min(x0, 0)
min(0, x0)
min(s(x0), s(x1))
twice(0)
twice(s(x0))
f(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.

(8) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


TWICE(s(x)) → TWICE(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TWICE(x1)  =  TWICE(x1)
s(x1)  =  s(x1)
-(x1, x2)  =  -(x1)
0  =  0
min(x1, x2)  =  min(x1)
twice(x1)  =  twice(x1)
f(x1, x2)  =  f

Lexicographic path order with status [LPO].
Quasi-Precedence:
[twice1, f] > [TWICE1, s1, -1, 0, min1]

Status:
TWICE1: [1]
s1: [1]
-1: [1]
0: []
min1: [1]
twice1: [1]
f: []


The following usable rules [FROCOS05] were oriented:

-(x, 0) → x
-(s(x), s(y)) → -(x, y)
min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
twice(0) → 0
twice(s(x)) → s(s(twice(x)))
f(s(x), s(y)) → f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) → f(-(x, min(x, y)), s(twice(min(x, y))))

(9) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

-(x, 0) → x
-(s(x), s(y)) → -(x, y)
min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
twice(0) → 0
twice(s(x)) → s(s(twice(x)))
f(s(x), s(y)) → f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) → f(-(x, min(x, y)), s(twice(min(x, y))))

The set Q consists of the following terms:

-(x0, 0)
-(s(x0), s(x1))
min(x0, 0)
min(0, x0)
min(s(x0), s(x1))
twice(0)
twice(s(x0))
f(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.

(10) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(11) TRUE

(12) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MIN(s(x), s(y)) → MIN(x, y)

The TRS R consists of the following rules:

-(x, 0) → x
-(s(x), s(y)) → -(x, y)
min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
twice(0) → 0
twice(s(x)) → s(s(twice(x)))
f(s(x), s(y)) → f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) → f(-(x, min(x, y)), s(twice(min(x, y))))

The set Q consists of the following terms:

-(x0, 0)
-(s(x0), s(x1))
min(x0, 0)
min(0, x0)
min(s(x0), s(x1))
twice(0)
twice(s(x0))
f(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.

(13) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MIN(s(x), s(y)) → MIN(x, y)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MIN(x1, x2)  =  MIN(x1, x2)
s(x1)  =  s(x1)
-(x1, x2)  =  -(x1)
0  =  0
min(x1, x2)  =  min(x1, x2)
twice(x1)  =  twice(x1)
f(x1, x2)  =  f

Lexicographic path order with status [LPO].
Quasi-Precedence:
0 > [MIN2, s1, -1]
[min2, twice1, f] > [MIN2, s1, -1]

Status:
MIN2: [1,2]
s1: [1]
-1: [1]
0: []
min2: [1,2]
twice1: [1]
f: []


The following usable rules [FROCOS05] were oriented:

-(x, 0) → x
-(s(x), s(y)) → -(x, y)
min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
twice(0) → 0
twice(s(x)) → s(s(twice(x)))
f(s(x), s(y)) → f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) → f(-(x, min(x, y)), s(twice(min(x, y))))

(14) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

-(x, 0) → x
-(s(x), s(y)) → -(x, y)
min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
twice(0) → 0
twice(s(x)) → s(s(twice(x)))
f(s(x), s(y)) → f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) → f(-(x, min(x, y)), s(twice(min(x, y))))

The set Q consists of the following terms:

-(x0, 0)
-(s(x0), s(x1))
min(x0, 0)
min(0, x0)
min(s(x0), s(x1))
twice(0)
twice(s(x0))
f(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.

(15) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(16) TRUE

(17) 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)
min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
twice(0) → 0
twice(s(x)) → s(s(twice(x)))
f(s(x), s(y)) → f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) → f(-(x, min(x, y)), s(twice(min(x, y))))

The set Q consists of the following terms:

-(x0, 0)
-(s(x0), s(x1))
min(x0, 0)
min(0, x0)
min(s(x0), s(x1))
twice(0)
twice(s(x0))
f(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.

(18) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


-1(s(x), s(y)) → -1(x, y)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
-1(x1, x2)  =  -1(x1, x2)
s(x1)  =  s(x1)
-(x1, x2)  =  -(x1)
0  =  0
min(x1, x2)  =  x1
twice(x1)  =  twice(x1)
f(x1, x2)  =  f

Lexicographic path order with status [LPO].
Quasi-Precedence:
twice1 > [s1, -1, f] > -^12 > 0

Status:
-^12: [1,2]
s1: [1]
-1: [1]
0: []
twice1: [1]
f: []


The following usable rules [FROCOS05] were oriented:

-(x, 0) → x
-(s(x), s(y)) → -(x, y)
min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
twice(0) → 0
twice(s(x)) → s(s(twice(x)))
f(s(x), s(y)) → f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) → f(-(x, min(x, y)), s(twice(min(x, y))))

(19) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

-(x, 0) → x
-(s(x), s(y)) → -(x, y)
min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
twice(0) → 0
twice(s(x)) → s(s(twice(x)))
f(s(x), s(y)) → f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) → f(-(x, min(x, y)), s(twice(min(x, y))))

The set Q consists of the following terms:

-(x0, 0)
-(s(x0), s(x1))
min(x0, 0)
min(0, x0)
min(s(x0), s(x1))
twice(0)
twice(s(x0))
f(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.

(20) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(21) TRUE

(22) Obligation:

Q DP problem:
The TRS P consists of the following rules:

F(s(x), s(y)) → F(-(x, min(x, y)), s(twice(min(x, y))))
F(s(x), s(y)) → F(-(y, min(x, y)), s(twice(min(x, y))))

The TRS R consists of the following rules:

-(x, 0) → x
-(s(x), s(y)) → -(x, y)
min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
twice(0) → 0
twice(s(x)) → s(s(twice(x)))
f(s(x), s(y)) → f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) → f(-(x, min(x, y)), s(twice(min(x, y))))

The set Q consists of the following terms:

-(x0, 0)
-(s(x0), s(x1))
min(x0, 0)
min(0, x0)
min(s(x0), s(x1))
twice(0)
twice(s(x0))
f(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.