(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
+(X, 0) → X
+(X, s(Y)) → s(+(X, Y))
double(X) → +(X, X)
f(0, s(0), X) → f(X, double(X), X)
g(X, Y) → X
g(X, Y) → Y
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(X, s(Y)) → +1(X, Y)
DOUBLE(X) → +1(X, X)
F(0, s(0), X) → F(X, double(X), X)
F(0, s(0), X) → DOUBLE(X)
The TRS R consists of the following rules:
+(X, 0) → X
+(X, s(Y)) → s(+(X, Y))
double(X) → +(X, X)
f(0, s(0), X) → f(X, double(X), X)
g(X, Y) → X
g(X, Y) → Y
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 2 SCCs with 2 less nodes.
(4) Complex Obligation (AND)
(5) 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
+(X, s(Y)) → s(+(X, Y))
double(X) → +(X, X)
f(0, s(0), X) → f(X, double(X), X)
g(X, Y) → X
g(X, Y) → Y
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(6) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
+1(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) =
x2
s(
x1) =
s(
x1)
+(
x1,
x2) =
+(
x1,
x2)
0 =
0
double(
x1) =
double(
x1)
f(
x1,
x2,
x3) =
x3
g(
x1,
x2) =
g(
x1,
x2)
Recursive Path Order [RPO].
Precedence:
[0, double1] > +2 > s1
g2 > s1
The following usable rules [FROCOS05] were oriented:
+(X, 0) → X
+(X, s(Y)) → s(+(X, Y))
double(X) → +(X, X)
f(0, s(0), X) → f(X, double(X), X)
g(X, Y) → X
g(X, Y) → Y
(7) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
+(X, 0) → X
+(X, s(Y)) → s(+(X, Y))
double(X) → +(X, X)
f(0, s(0), X) → f(X, double(X), X)
g(X, Y) → X
g(X, Y) → Y
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(8) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(9) TRUE
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(0, s(0), X) → F(X, double(X), X)
The TRS R consists of the following rules:
+(X, 0) → X
+(X, s(Y)) → s(+(X, Y))
double(X) → +(X, X)
f(0, s(0), X) → f(X, double(X), X)
g(X, Y) → X
g(X, Y) → Y
Q is empty.
We have to consider all minimal (P,Q,R)-chains.