(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
div(X, e) → i(X)
i(div(X, Y)) → div(Y, X)
div(div(X, Y), Z) → div(Y, div(i(X), 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:
DIV(X, e) → I(X)
I(div(X, Y)) → DIV(Y, X)
DIV(div(X, Y), Z) → DIV(Y, div(i(X), Z))
DIV(div(X, Y), Z) → DIV(i(X), Z)
DIV(div(X, Y), Z) → I(X)
The TRS R consists of the following rules:
div(X, e) → i(X)
i(div(X, Y)) → div(Y, X)
div(div(X, Y), Z) → div(Y, div(i(X), Z))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
DIV(X, e) → I(X)
I(div(X, Y)) → DIV(Y, X)
DIV(div(X, Y), Z) → DIV(Y, div(i(X), Z))
DIV(div(X, Y), Z) → DIV(i(X), Z)
DIV(div(X, Y), Z) → I(X)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
DIV(
x0,
x1,
x2) =
DIV(
x0,
x1)
I(
x0,
x1) =
I(
x1)
Tags:
DIV has argument tags [2,0,3] and root tag 0
I has argument tags [4,2] and root tag 1
Comparison: MS
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
DIV(
x1,
x2) =
x1
e =
e
I(
x1) =
I
div(
x1,
x2) =
div(
x1,
x2)
i(
x1) =
i(
x1)
Recursive path order with status [RPO].
Quasi-Precedence:
[div2, i1] > [e, I]
Status:
e: multiset
I: multiset
div2: [1,2]
i1: [1]
The following usable rules [FROCOS05] were oriented:
i(div(X, Y)) → div(Y, X)
div(X, e) → i(X)
div(div(X, Y), Z) → div(Y, div(i(X), Z))
(4) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
div(X, e) → i(X)
i(div(X, Y)) → div(Y, X)
div(div(X, Y), Z) → div(Y, div(i(X), Z))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(6) TRUE