(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
active(f(a, b, X)) → mark(f(X, X, X))
active(c) → mark(a)
active(c) → mark(b)
active(f(X1, X2, X3)) → f(X1, X2, active(X3))
f(X1, X2, mark(X3)) → mark(f(X1, X2, X3))
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3))
proper(a) → ok(a)
proper(b) → ok(b)
proper(c) → ok(c)
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3))
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(a, b, X)) → mark(f(X, X, X))
active(c) → mark(a)
active(c) → mark(b)
active(f(X1, X2, X3)) → f(X1, X2, active(X3))
f(X1, X2, mark(X3)) → mark(f(X1, X2, X3))
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3))
proper(a) → ok(a)
proper(b) → ok(b)
proper(c) → ok(c)
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3))
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: {3}
a: empty set
b: empty set
c: 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(a, b, X) → f(X, X, X)
c → a
c → b
The replacement map contains the following entries:
f: {3}
a: empty set
b: empty set
c: empty set
(3) PoloCSRProof (EQUIVALENT transformation)
The following rules can be removed because they are oriented strictly by a µ-monotonic polynomial ordering:
c → a
c → b
Used ordering:
Polynomial interpretation [POLO]:
POL(a) = 1
POL(b) = 1
POL(c) = 2
POL(f(x1, x2, x3)) = x3
(4) Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:
f(a, b, X) → f(X, X, X)
The replacement map contains the following entries:
f: {3}
a: empty set
b: empty set
(5) CSRInnermostProof (EQUIVALENT transformation)
The CSR is orthogonal. By [CS_Inn] we can switch to innermost.
(6) Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:
f(a, b, X) → f(X, X, X)
The replacement map contains the following entries:
f: {3}
a: empty set
b: empty set
Innermost Strategy.
(7) CSDependencyPairsProof (EQUIVALENT transformation)
Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
(8) Obligation:
Q-restricted context-sensitive dependency pair problem:
For all symbols f in {
f,
F} we have µ(f) = {3}.
The ordinary context-sensitive dependency pairs DP
o are:
F(a, b, X) → F(X, X, X)
The TRS R consists of the following rules:
f(a, b, X) → f(X, X, X)
The set Q consists of the following terms:
f(a, b, x0)
(9) QCSDependencyGraphProof (EQUIVALENT transformation)
The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs.
The rules F(a, b, z0) → F(z0, z0, z0) and F(a, b, x0) → F(x0, x0, x0) form no chain, because ECapµ(F(z0, z0, z0)) = F(z0, z0, z0) does not unify with F(a, b, x0).
(10) TRUE