(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
times(x, plus(y, 1)) → plus(times(x, plus(y, times(1, 0))), x)
times(x, 1) → x
plus(x, 0) → x
times(x, 0) → 0
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:
times(x, plus(y, 1)) → plus(times(x, plus(y, times(1, 0))), x)
times(x, 1) → x
plus(x, 0) → x
times(x, 0) → 0
The set Q consists of the following terms:
times(x0, plus(x1, 1))
times(x0, 1)
plus(x0, 0)
times(x0, 0)
(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:
TIMES(x, plus(y, 1)) → PLUS(times(x, plus(y, times(1, 0))), x)
TIMES(x, plus(y, 1)) → TIMES(x, plus(y, times(1, 0)))
TIMES(x, plus(y, 1)) → PLUS(y, times(1, 0))
TIMES(x, plus(y, 1)) → TIMES(1, 0)
The TRS R consists of the following rules:
times(x, plus(y, 1)) → plus(times(x, plus(y, times(1, 0))), x)
times(x, 1) → x
plus(x, 0) → x
times(x, 0) → 0
The set Q consists of the following terms:
times(x0, plus(x1, 1))
times(x0, 1)
plus(x0, 0)
times(x0, 0)
We have to consider all minimal (P,Q,R)-chains.
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
TIMES(x, plus(y, 1)) → TIMES(x, plus(y, times(1, 0)))
The TRS R consists of the following rules:
times(x, plus(y, 1)) → plus(times(x, plus(y, times(1, 0))), x)
times(x, 1) → x
plus(x, 0) → x
times(x, 0) → 0
The set Q consists of the following terms:
times(x0, plus(x1, 1))
times(x0, 1)
plus(x0, 0)
times(x0, 0)
We have to consider all minimal (P,Q,R)-chains.
(7) 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.
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
TIMES(x, plus(y, 1)) → TIMES(x, plus(y, times(1, 0)))
The TRS R consists of the following rules:
times(x, 0) → 0
plus(x, 0) → x
The set Q consists of the following terms:
times(x0, plus(x1, 1))
times(x0, 1)
plus(x0, 0)
times(x0, 0)
We have to consider all minimal (P,Q,R)-chains.
(9) MRRProof (EQUIVALENT transformation)
By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented rules of the TRS R:
plus(x, 0) → x
Used ordering: Polynomial interpretation [POLO]:
POL(0) = 0
POL(1) = 0
POL(TIMES(x1, x2)) = x1 + x2
POL(plus(x1, x2)) = 1 + x1 + x2
POL(times(x1, x2)) = x1 + x2
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
TIMES(x, plus(y, 1)) → TIMES(x, plus(y, times(1, 0)))
The TRS R consists of the following rules:
times(x, 0) → 0
The set Q consists of the following terms:
times(x0, plus(x1, 1))
times(x0, 1)
plus(x0, 0)
times(x0, 0)
We have to consider all minimal (P,Q,R)-chains.
(11) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
(12) TRUE