(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(0) → 1
f(s(x)) → g(x, s(x))
g(0, y) → y
g(s(x), y) → g(x, +(y, s(x)))
+(x, 0) → x
+(x, s(y)) → s(+(x, y))
g(s(x), y) → g(x, s(+(y, 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:
f(0) → 1
f(s(x)) → g(x, s(x))
g(0, y) → y
g(s(x), y) → g(x, +(y, s(x)))
+(x, 0) → x
+(x, s(y)) → s(+(x, y))
g(s(x), y) → g(x, s(+(y, x)))
The set Q consists of the following terms:
f(0)
f(s(x0))
g(0, x0)
g(s(x0), x1)
+(x0, 0)
+(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:
F(s(x)) → G(x, s(x))
G(s(x), y) → G(x, +(y, s(x)))
G(s(x), y) → +1(y, s(x))
+1(x, s(y)) → +1(x, y)
G(s(x), y) → G(x, s(+(y, x)))
G(s(x), y) → +1(y, x)
The TRS R consists of the following rules:
f(0) → 1
f(s(x)) → g(x, s(x))
g(0, y) → y
g(s(x), y) → g(x, +(y, s(x)))
+(x, 0) → x
+(x, s(y)) → s(+(x, y))
g(s(x), y) → g(x, s(+(y, x)))
The set Q consists of the following terms:
f(0)
f(s(x0))
g(0, x0)
g(s(x0), x1)
+(x0, 0)
+(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 2 SCCs with 3 less nodes.
(6) Complex Obligation (AND)
(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:
f(0) → 1
f(s(x)) → g(x, s(x))
g(0, y) → y
g(s(x), y) → g(x, +(y, s(x)))
+(x, 0) → x
+(x, s(y)) → s(+(x, y))
g(s(x), y) → g(x, s(+(y, x)))
The set Q consists of the following terms:
f(0)
f(s(x0))
g(0, x0)
g(s(x0), x1)
+(x0, 0)
+(x0, s(x1))
We have to consider all minimal (P,Q,R)-chains.
(8) UsableRulesProof (EQUIVALENT transformation)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
(9) Obligation:
Q DP problem:
The TRS P consists of the following rules:
+1(x, s(y)) → +1(x, y)
R is empty.
The set Q consists of the following terms:
f(0)
f(s(x0))
g(0, x0)
g(s(x0), x1)
+(x0, 0)
+(x0, s(x1))
We have to consider all minimal (P,Q,R)-chains.
(10) QReductionProof (EQUIVALENT transformation)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
f(0)
f(s(x0))
g(0, x0)
g(s(x0), x1)
+(x0, 0)
+(x0, s(x1))
(11) Obligation:
Q DP problem:
The TRS P consists of the following rules:
+1(x, s(y)) → +1(x, y)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(12) QDPSizeChangeProof (EQUIVALENT transformation)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.
From the DPs we obtained the following set of size-change graphs:
- +1(x, s(y)) → +1(x, y)
The graph contains the following edges 1 >= 1, 2 > 2
(13) TRUE
(14) Obligation:
Q DP problem:
The TRS P consists of the following rules:
G(s(x), y) → G(x, s(+(y, x)))
G(s(x), y) → G(x, +(y, s(x)))
The TRS R consists of the following rules:
f(0) → 1
f(s(x)) → g(x, s(x))
g(0, y) → y
g(s(x), y) → g(x, +(y, s(x)))
+(x, 0) → x
+(x, s(y)) → s(+(x, y))
g(s(x), y) → g(x, s(+(y, x)))
The set Q consists of the following terms:
f(0)
f(s(x0))
g(0, x0)
g(s(x0), x1)
+(x0, 0)
+(x0, s(x1))
We have to consider all minimal (P,Q,R)-chains.
(15) UsableRulesProof (EQUIVALENT transformation)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
(16) Obligation:
Q DP problem:
The TRS P consists of the following rules:
G(s(x), y) → G(x, s(+(y, x)))
G(s(x), y) → G(x, +(y, s(x)))
The TRS R consists of the following rules:
+(x, s(y)) → s(+(x, y))
+(x, 0) → x
The set Q consists of the following terms:
f(0)
f(s(x0))
g(0, x0)
g(s(x0), x1)
+(x0, 0)
+(x0, s(x1))
We have to consider all minimal (P,Q,R)-chains.
(17) QReductionProof (EQUIVALENT transformation)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
f(0)
f(s(x0))
g(0, x0)
g(s(x0), x1)
(18) Obligation:
Q DP problem:
The TRS P consists of the following rules:
G(s(x), y) → G(x, s(+(y, x)))
G(s(x), y) → G(x, +(y, s(x)))
The TRS R consists of the following rules:
+(x, s(y)) → s(+(x, y))
+(x, 0) → x
The set Q consists of the following terms:
+(x0, 0)
+(x0, s(x1))
We have to consider all minimal (P,Q,R)-chains.
(19) QDPSizeChangeProof (EQUIVALENT transformation)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.
From the DPs we obtained the following set of size-change graphs:
- G(s(x), y) → G(x, s(+(y, x)))
The graph contains the following edges 1 > 1
- G(s(x), y) → G(x, +(y, s(x)))
The graph contains the following edges 1 > 1
(20) TRUE