(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))
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,
x1)
IF(
x0,
x1,
x2,
x3) =
IF(
x0,
x1,
x2)
ACTIVATE(
x0,
x1) =
ACTIVATE(
x0,
x1)
Tags:
F has argument tags [4,1] and root tag 0
IF has argument tags [1,0,0,4] and root tag 1
ACTIVATE has argument tags [0,1] and root tag 1
Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
F(
x1) =
F
IF(
x1,
x2,
x3) =
x3
c =
c
n__f(
x1) =
x1
true =
true
false =
false
ACTIVATE(
x1) =
ACTIVATE
Recursive path order with status [RPO].
Quasi-Precedence:
[false, ACTIVATE] > [F, c, true]
Status:
F: []
c: multiset
true: multiset
false: multiset
ACTIVATE: multiset
The following usable rules [FROCOS05] were oriented:
none
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
IF(false, X, Y) → ACTIVATE(Y)
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) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
(6) TRUE