(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
active(f(X, X)) → mark(f(a, b))
active(b) → mark(a)
active(f(X1, X2)) → f(active(X1), X2)
f(mark(X1), X2) → mark(f(X1, X2))
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(a) → ok(a)
proper(b) → ok(b)
f(ok(X1), ok(X2)) → ok(f(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
(1) QTRSToCSRProof (EQUIVALENT transformation)
The following Q TRS is given: Q restricted rewrite system:
The TRS R consists of the following rules:
active(f(X, X)) → mark(f(a, b))
active(b) → mark(a)
active(f(X1, X2)) → f(active(X1), X2)
f(mark(X1), X2) → mark(f(X1, X2))
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(a) → ok(a)
proper(b) → ok(b)
f(ok(X1), ok(X2)) → ok(f(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
Special symbols used for the transformation (see [GM04]):
top:
top, active:
active, mark:
mark, ok:
ok, proper:
properThe replacement map contains the following entries:
f: {1}
a: empty set
b: empty set
The QTRS contained all rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is complete (and sound).
(2) Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:
f(X, X) → f(a, b)
b → a
The replacement map contains the following entries:
f: {1}
a: empty set
b: empty set
(3) PoloCSRProof (EQUIVALENT transformation)
The following rules can be removed because they are oriented strictly by a µ-monotonic polynomial ordering:
b → a
Used ordering:
Polynomial interpretation [POLO]:
POL(a) = 0
POL(b) = 2
POL(f(x1, x2)) = x1
(4) Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:
f(X, X) → f(a, b)
The replacement map contains the following entries:
f: {1}
a: empty set
b: empty set
(5) CSDependencyPairsProof (EQUIVALENT transformation)
Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
(6) Obligation:
Q-restricted context-sensitive dependency pair problem:
For all symbols f in {
f,
F} we have µ(f) = {1}.
The ordinary context-sensitive dependency pairs DP
o are:
F(X, X) → F(a, b)
The TRS R consists of the following rules:
f(X, X) → f(a, b)
Q is empty.
(7) QCSDependencyGraphProof (EQUIVALENT transformation)
The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs.
The rules F(z0, z0) → F(a, b) and F(x0, x0) → F(a, b) form no chain, because ECapµ(F(a, b)) = F(a, b) does not unify with F(x0, x0).
(8) TRUE