(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(b(x)) → b(a(a(x)))
b(c(x)) → c(b(b(x)))
c(a(x)) → a(c(c(x)))
u(a(x)) → x
v(b(x)) → x
w(c(x)) → x
a(u(x)) → x
b(v(x)) → x
c(w(x)) → x
Q is empty.
(1) QTRS Reverse (EQUIVALENT transformation)
We applied the QTRS Reverse Processor [REVERSE].
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
b(a(x)) → a(a(b(x)))
c(b(x)) → b(b(c(x)))
a(c(x)) → c(c(a(x)))
a(u(x)) → x
b(v(x)) → x
c(w(x)) → x
u(a(x)) → x
v(b(x)) → x
w(c(x)) → x
Q is empty.
(3) QTRS Reverse (EQUIVALENT transformation)
We applied the QTRS Reverse Processor [REVERSE].
(4) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
b(a(x)) → a(a(b(x)))
c(b(x)) → b(b(c(x)))
a(c(x)) → c(c(a(x)))
a(u(x)) → x
b(v(x)) → x
c(w(x)) → x
u(a(x)) → x
v(b(x)) → x
w(c(x)) → x
Q is empty.
(5) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(a(x1)) = x1
POL(b(x1)) = x1
POL(c(x1)) = x1
POL(u(x1)) = 1 + x1
POL(v(x1)) = x1
POL(w(x1)) = x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
u(a(x)) → x
a(u(x)) → x
(6) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(b(x)) → b(a(a(x)))
b(c(x)) → c(b(b(x)))
c(a(x)) → a(c(c(x)))
v(b(x)) → x
w(c(x)) → x
b(v(x)) → x
c(w(x)) → x
Q is empty.
(7) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(a(x1)) = x1
POL(b(x1)) = x1
POL(c(x1)) = x1
POL(v(x1)) = 1 + x1
POL(w(x1)) = x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
v(b(x)) → x
b(v(x)) → x
(8) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(b(x)) → b(a(a(x)))
b(c(x)) → c(b(b(x)))
c(a(x)) → a(c(c(x)))
w(c(x)) → x
c(w(x)) → x
Q is empty.
(9) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(a(x1)) = x1
POL(b(x1)) = x1
POL(c(x1)) = x1
POL(w(x1)) = 1 + x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
w(c(x)) → x
c(w(x)) → x
(10) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(b(x)) → b(a(a(x)))
b(c(x)) → c(b(b(x)))
c(a(x)) → a(c(c(x)))
Q is empty.
(11) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(12) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A(b(x)) → B(a(a(x)))
A(b(x)) → A(a(x))
A(b(x)) → A(x)
B(c(x)) → C(b(b(x)))
B(c(x)) → B(b(x))
B(c(x)) → B(x)
C(a(x)) → A(c(c(x)))
C(a(x)) → C(c(x))
C(a(x)) → C(x)
The TRS R consists of the following rules:
a(b(x)) → b(a(a(x)))
b(c(x)) → c(b(b(x)))
c(a(x)) → a(c(c(x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(13) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
C(a(x)) → A(c(c(x)))
C(a(x)) → C(c(x))
C(a(x)) → C(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( A(x1) ) = 1
POL( B(x1) ) = 1
POL( C(x1) ) = x1 + 1
POL( b(x1) ) = max{0, -1}
POL( c(x1) ) = x1
POL( a(x1) ) = x1 + 1
The following usable rules [FROCOS05] were oriented:
b(c(x)) → c(b(b(x)))
a(b(x)) → b(a(a(x)))
c(a(x)) → a(c(c(x)))
(14) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A(b(x)) → B(a(a(x)))
A(b(x)) → A(a(x))
A(b(x)) → A(x)
B(c(x)) → C(b(b(x)))
B(c(x)) → B(b(x))
B(c(x)) → B(x)
The TRS R consists of the following rules:
a(b(x)) → b(a(a(x)))
b(c(x)) → c(b(b(x)))
c(a(x)) → a(c(c(x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(15) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
B(c(x)) → C(b(b(x)))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
The following usable rules [FROCOS05] were oriented:
b(c(x)) → c(b(b(x)))
a(b(x)) → b(a(a(x)))
c(a(x)) → a(c(c(x)))
(16) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A(b(x)) → B(a(a(x)))
A(b(x)) → A(a(x))
A(b(x)) → A(x)
B(c(x)) → B(b(x))
B(c(x)) → B(x)
C(a(x)) → A(c(c(x)))
C(a(x)) → C(c(x))
C(a(x)) → C(x)
The TRS R consists of the following rules:
a(b(x)) → b(a(a(x)))
b(c(x)) → c(b(b(x)))
c(a(x)) → a(c(c(x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(17) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 2 less nodes.
(18) Complex Obligation (AND)
(19) Obligation:
Q DP problem:
The TRS P consists of the following rules:
B(c(x)) → B(x)
B(c(x)) → B(b(x))
The TRS R consists of the following rules:
a(b(x)) → b(a(a(x)))
b(c(x)) → c(b(b(x)))
c(a(x)) → a(c(c(x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(20) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
B(c(x)) → B(x)
B(c(x)) → B(b(x))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( B(x1) ) = x1 + 1
POL( b(x1) ) = x1
POL( c(x1) ) = x1 + 1
POL( a(x1) ) = 1
The following usable rules [FROCOS05] were oriented:
b(c(x)) → c(b(b(x)))
a(b(x)) → b(a(a(x)))
c(a(x)) → a(c(c(x)))
(21) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
a(b(x)) → b(a(a(x)))
b(c(x)) → c(b(b(x)))
c(a(x)) → a(c(c(x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(22) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
B(c(x)) → B(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
The following usable rules [FROCOS05] were oriented:
b(c(x)) → c(b(b(x)))
a(b(x)) → b(a(a(x)))
c(a(x)) → a(c(c(x)))
(23) Obligation:
Q DP problem:
The TRS P consists of the following rules:
B(c(x)) → B(b(x))
The TRS R consists of the following rules:
a(b(x)) → b(a(a(x)))
b(c(x)) → c(b(b(x)))
c(a(x)) → a(c(c(x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(24) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
B(c(x)) → B(b(x))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( B(x1) ) = x1 + 1
POL( b(x1) ) = x1
POL( c(x1) ) = x1 + 1
POL( a(x1) ) = max{0, -1}
The following usable rules [FROCOS05] were oriented:
b(c(x)) → c(b(b(x)))
a(b(x)) → b(a(a(x)))
c(a(x)) → a(c(c(x)))
(25) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
a(b(x)) → b(a(a(x)))
b(c(x)) → c(b(b(x)))
c(a(x)) → a(c(c(x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(26) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(27) TRUE
(28) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A(b(x)) → A(x)
A(b(x)) → A(a(x))
The TRS R consists of the following rules:
a(b(x)) → b(a(a(x)))
b(c(x)) → c(b(b(x)))
c(a(x)) → a(c(c(x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(29) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
A(b(x)) → A(x)
A(b(x)) → A(a(x))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( A(x1) ) = x1 + 1
POL( b(x1) ) = x1 + 1
POL( c(x1) ) = 1
POL( a(x1) ) = x1
The following usable rules [FROCOS05] were oriented:
b(c(x)) → c(b(b(x)))
a(b(x)) → b(a(a(x)))
c(a(x)) → a(c(c(x)))
(30) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
a(b(x)) → b(a(a(x)))
b(c(x)) → c(b(b(x)))
c(a(x)) → a(c(c(x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(31) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(32) TRUE
(33) Obligation:
Q DP problem:
The TRS P consists of the following rules:
C(a(x)) → C(x)
C(a(x)) → C(c(x))
The TRS R consists of the following rules:
a(b(x)) → b(a(a(x)))
b(c(x)) → c(b(b(x)))
c(a(x)) → a(c(c(x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(34) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
C(a(x)) → C(x)
C(a(x)) → C(c(x))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( C(x1) ) = x1 + 1
POL( b(x1) ) = 1
POL( c(x1) ) = x1
POL( a(x1) ) = x1 + 1
The following usable rules [FROCOS05] were oriented:
b(c(x)) → c(b(b(x)))
a(b(x)) → b(a(a(x)))
c(a(x)) → a(c(c(x)))
(35) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
a(b(x)) → b(a(a(x)))
b(c(x)) → c(b(b(x)))
c(a(x)) → a(c(c(x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(36) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(37) TRUE