(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(X) → if(X, c, n__f(true))
if(true, X, Y) → X
if(false, X, Y) → activate(Y)
f(X) → n__f(X)
activate(n__f(X)) → f(X)
activate(X) → X
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:
F(X) → IF(X, c, n__f(true))
IF(false, X, Y) → ACTIVATE(Y)
ACTIVATE(n__f(X)) → F(X)
The TRS R consists of the following rules:
f(X) → if(X, c, n__f(true))
if(true, X, Y) → X
if(false, X, Y) → activate(Y)
f(X) → n__f(X)
activate(n__f(X)) → f(X)
activate(X) → X
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.
F(X) → IF(X, c, n__f(true))
IF(false, X, Y) → ACTIVATE(Y)
ACTIVATE(n__f(X)) → F(X)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
F(
x0,
x1) =
F(
x0)
IF(
x0,
x1,
x2,
x3) =
IF(
x0,
x1,
x3)
ACTIVATE(
x0,
x1) =
ACTIVATE(
x0)
Tags:
F has argument tags [3,2] and root tag 3
IF has argument tags [0,3,4,3] and root tag 1
ACTIVATE has argument tags [7,1] and root tag 2
Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:
POL(ACTIVATE(x1)) = x1
POL(F(x1)) = x1
POL(IF(x1, x2, x3)) = x1 + x3
POL(c) = 1
POL(false) = 1
POL(n__f(x1)) = x1
POL(true) = 0
The following usable rules [FROCOS05] were oriented:
none
(4) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
f(X) → if(X, c, n__f(true))
if(true, X, Y) → X
if(false, X, Y) → activate(Y)
f(X) → n__f(X)
activate(n__f(X)) → f(X)
activate(X) → X
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