Termination w.r.t. Q of the following Term Rewriting System could not be shown:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__f(a, b, X) → a__f(mark(X), X, mark(X))
a__c → a
a__c → b
mark(f(X1, X2, X3)) → a__f(mark(X1), X2, mark(X3))
mark(c) → a__c
mark(a) → a
mark(b) → b
a__f(X1, X2, X3) → f(X1, X2, X3)
a__c → c
Q is empty.
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
a__f(a, b, X) → a__f(mark(X), X, mark(X))
a__c → a
a__c → b
mark(f(X1, X2, X3)) → a__f(mark(X1), X2, mark(X3))
mark(c) → a__c
mark(a) → a
mark(b) → b
a__f(X1, X2, X3) → f(X1, X2, X3)
a__c → c
Q is empty.
Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:
MARK(f(X1, X2, X3)) → MARK(X3)
A__F(a, b, X) → MARK(X)
MARK(c) → A__C
MARK(f(X1, X2, X3)) → MARK(X1)
MARK(f(X1, X2, X3)) → A__F(mark(X1), X2, mark(X3))
A__F(a, b, X) → A__F(mark(X), X, mark(X))
The TRS R consists of the following rules:
a__f(a, b, X) → a__f(mark(X), X, mark(X))
a__c → a
a__c → b
mark(f(X1, X2, X3)) → a__f(mark(X1), X2, mark(X3))
mark(c) → a__c
mark(a) → a
mark(b) → b
a__f(X1, X2, X3) → f(X1, X2, X3)
a__c → c
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
MARK(f(X1, X2, X3)) → MARK(X3)
A__F(a, b, X) → MARK(X)
MARK(c) → A__C
MARK(f(X1, X2, X3)) → MARK(X1)
MARK(f(X1, X2, X3)) → A__F(mark(X1), X2, mark(X3))
A__F(a, b, X) → A__F(mark(X), X, mark(X))
The TRS R consists of the following rules:
a__f(a, b, X) → a__f(mark(X), X, mark(X))
a__c → a
a__c → b
mark(f(X1, X2, X3)) → a__f(mark(X1), X2, mark(X3))
mark(c) → a__c
mark(a) → a
mark(b) → b
a__f(X1, X2, X3) → f(X1, X2, X3)
a__c → c
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
MARK(f(X1, X2, X3)) → MARK(X3)
A__F(a, b, X) → MARK(X)
MARK(f(X1, X2, X3)) → MARK(X1)
A__F(a, b, X) → A__F(mark(X), X, mark(X))
MARK(f(X1, X2, X3)) → A__F(mark(X1), X2, mark(X3))
The TRS R consists of the following rules:
a__f(a, b, X) → a__f(mark(X), X, mark(X))
a__c → a
a__c → b
mark(f(X1, X2, X3)) → a__f(mark(X1), X2, mark(X3))
mark(c) → a__c
mark(a) → a
mark(b) → b
a__f(X1, X2, X3) → f(X1, X2, X3)
a__c → c
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(f(X1, X2, X3)) → A__F(mark(X1), X2, mark(X3)) at position [0] we obtained the following new rules:
MARK(f(a, y1, y2)) → A__F(a, y1, mark(y2))
MARK(f(f(x0, x1, x2), y1, y2)) → A__F(a__f(mark(x0), x1, mark(x2)), y1, mark(y2))
MARK(f(c, y1, y2)) → A__F(a__c, y1, mark(y2))
MARK(f(b, y1, y2)) → A__F(b, y1, mark(y2))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
MARK(f(X1, X2, X3)) → MARK(X3)
MARK(f(f(x0, x1, x2), y1, y2)) → A__F(a__f(mark(x0), x1, mark(x2)), y1, mark(y2))
MARK(f(a, y1, y2)) → A__F(a, y1, mark(y2))
A__F(a, b, X) → MARK(X)
MARK(f(b, y1, y2)) → A__F(b, y1, mark(y2))
MARK(f(c, y1, y2)) → A__F(a__c, y1, mark(y2))
A__F(a, b, X) → A__F(mark(X), X, mark(X))
MARK(f(X1, X2, X3)) → MARK(X1)
The TRS R consists of the following rules:
a__f(a, b, X) → a__f(mark(X), X, mark(X))
a__c → a
a__c → b
mark(f(X1, X2, X3)) → a__f(mark(X1), X2, mark(X3))
mark(c) → a__c
mark(a) → a
mark(b) → b
a__f(X1, X2, X3) → f(X1, X2, X3)
a__c → c
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
MARK(f(X1, X2, X3)) → MARK(X3)
MARK(f(f(x0, x1, x2), y1, y2)) → A__F(a__f(mark(x0), x1, mark(x2)), y1, mark(y2))
MARK(f(a, y1, y2)) → A__F(a, y1, mark(y2))
A__F(a, b, X) → MARK(X)
MARK(f(c, y1, y2)) → A__F(a__c, y1, mark(y2))
A__F(a, b, X) → A__F(mark(X), X, mark(X))
MARK(f(X1, X2, X3)) → MARK(X1)
The TRS R consists of the following rules:
a__f(a, b, X) → a__f(mark(X), X, mark(X))
a__c → a
a__c → b
mark(f(X1, X2, X3)) → a__f(mark(X1), X2, mark(X3))
mark(c) → a__c
mark(a) → a
mark(b) → b
a__f(X1, X2, X3) → f(X1, X2, X3)
a__c → c
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(f(c, y1, y2)) → A__F(a__c, y1, mark(y2)) at position [0] we obtained the following new rules:
MARK(f(c, y0, y1)) → A__F(b, y0, mark(y1))
MARK(f(c, y0, y1)) → A__F(c, y0, mark(y1))
MARK(f(c, y0, y1)) → A__F(a, y0, mark(y1))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
MARK(f(c, y0, y1)) → A__F(b, y0, mark(y1))
MARK(f(X1, X2, X3)) → MARK(X3)
MARK(f(a, y1, y2)) → A__F(a, y1, mark(y2))
MARK(f(f(x0, x1, x2), y1, y2)) → A__F(a__f(mark(x0), x1, mark(x2)), y1, mark(y2))
A__F(a, b, X) → MARK(X)
MARK(f(c, y0, y1)) → A__F(c, y0, mark(y1))
MARK(f(X1, X2, X3)) → MARK(X1)
A__F(a, b, X) → A__F(mark(X), X, mark(X))
MARK(f(c, y0, y1)) → A__F(a, y0, mark(y1))
The TRS R consists of the following rules:
a__f(a, b, X) → a__f(mark(X), X, mark(X))
a__c → a
a__c → b
mark(f(X1, X2, X3)) → a__f(mark(X1), X2, mark(X3))
mark(c) → a__c
mark(a) → a
mark(b) → b
a__f(X1, X2, X3) → f(X1, X2, X3)
a__c → c
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
MARK(f(X1, X2, X3)) → MARK(X3)
MARK(f(f(x0, x1, x2), y1, y2)) → A__F(a__f(mark(x0), x1, mark(x2)), y1, mark(y2))
MARK(f(a, y1, y2)) → A__F(a, y1, mark(y2))
A__F(a, b, X) → MARK(X)
A__F(a, b, X) → A__F(mark(X), X, mark(X))
MARK(f(X1, X2, X3)) → MARK(X1)
MARK(f(c, y0, y1)) → A__F(a, y0, mark(y1))
The TRS R consists of the following rules:
a__f(a, b, X) → a__f(mark(X), X, mark(X))
a__c → a
a__c → b
mark(f(X1, X2, X3)) → a__f(mark(X1), X2, mark(X3))
mark(c) → a__c
mark(a) → a
mark(b) → b
a__f(X1, X2, X3) → f(X1, X2, X3)
a__c → c
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
MARK(f(f(x0, x1, x2), y1, y2)) → A__F(a__f(mark(x0), x1, mark(x2)), y1, mark(y2))
The remaining pairs can at least be oriented weakly.
MARK(f(X1, X2, X3)) → MARK(X3)
MARK(f(a, y1, y2)) → A__F(a, y1, mark(y2))
A__F(a, b, X) → MARK(X)
A__F(a, b, X) → A__F(mark(X), X, mark(X))
MARK(f(X1, X2, X3)) → MARK(X1)
MARK(f(c, y0, y1)) → A__F(a, y0, mark(y1))
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( f(x1, ..., x3) ) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
M( a__f(x1, ..., x3) ) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
Tuple symbols:
M( A__F(x1, ..., x3) ) = | 0 | + | | · | x1 | + | | · | x2 | + | | · | x3 |
Matrix type:
We used a basic matrix type which is not further parametrizeable.
As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented:
a__f(a, b, X) → a__f(mark(X), X, mark(X))
a__c → b
a__c → a
mark(c) → a__c
mark(f(X1, X2, X3)) → a__f(mark(X1), X2, mark(X3))
mark(b) → b
mark(a) → a
a__c → c
a__f(X1, X2, X3) → f(X1, X2, X3)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
MARK(f(X1, X2, X3)) → MARK(X3)
MARK(f(a, y1, y2)) → A__F(a, y1, mark(y2))
A__F(a, b, X) → MARK(X)
MARK(f(X1, X2, X3)) → MARK(X1)
A__F(a, b, X) → A__F(mark(X), X, mark(X))
MARK(f(c, y0, y1)) → A__F(a, y0, mark(y1))
The TRS R consists of the following rules:
a__f(a, b, X) → a__f(mark(X), X, mark(X))
a__c → a
a__c → b
mark(f(X1, X2, X3)) → a__f(mark(X1), X2, mark(X3))
mark(c) → a__c
mark(a) → a
mark(b) → b
a__f(X1, X2, X3) → f(X1, X2, X3)
a__c → c
Q is empty.
We have to consider all minimal (P,Q,R)-chains.