Term Rewriting System R:
[X, Y]
f(X) -> if(X, c, nf(ntrue))
f(X) -> nf(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> ntrue
activate(nf(X)) -> f(activate(X))
activate(ntrue) -> true
activate(X) -> X

Innermost Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

F(X) -> IF(X, c, nf(ntrue))
IF(false, X, Y) -> ACTIVATE(Y)
ACTIVATE(nf(X)) -> F(activate(X))
ACTIVATE(nf(X)) -> ACTIVATE(X)
ACTIVATE(ntrue) -> TRUE

Furthermore, R contains one SCC.

R
DPs
→DP Problem 1
Argument Filtering and Ordering

Dependency Pairs:

ACTIVATE(nf(X)) -> ACTIVATE(X)
ACTIVATE(nf(X)) -> F(activate(X))
IF(false, X, Y) -> ACTIVATE(Y)
F(X) -> IF(X, c, nf(ntrue))

Rules:

f(X) -> if(X, c, nf(ntrue))
f(X) -> nf(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> ntrue
activate(nf(X)) -> f(activate(X))
activate(ntrue) -> true
activate(X) -> X

Strategy:

innermost

The following dependency pair can be strictly oriented:

ACTIVATE(nf(X)) -> F(activate(X))

The following usable rules for innermost w.r.t. to the AFS can be oriented:

activate(nf(X)) -> f(activate(X))
activate(ntrue) -> true
activate(X) -> X
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
f(X) -> if(X, c, nf(ntrue))
f(X) -> nf(X)
true -> ntrue

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(n__f(x1)) =  x1 POL(activate(x1)) =  x1 POL(n__true) =  0 POL(c) =  0 POL(if(x1, x2, x3)) =  x1 + x2 + x3 POL(false) =  1 POL(true) =  0 POL(ACTIVATE(x1)) =  1 + x1 POL(F(x1)) =  x1 POL(IF(x1, x2, x3)) =  x1 + x2 + x3 POL(f(x1)) =  x1

resulting in one new DP problem.
Used Argument Filtering System:
ACTIVATE(x1) -> ACTIVATE(x1)
F(x1) -> F(x1)
nf(x1) -> nf(x1)
activate(x1) -> activate(x1)
IF(x1, x2, x3) -> IF(x1, x2, x3)
f(x1) -> f(x1)
true -> true
if(x1, x2, x3) -> if(x1, x2, x3)

R
DPs
→DP Problem 1
AFS
→DP Problem 2
Dependency Graph

Dependency Pairs:

ACTIVATE(nf(X)) -> ACTIVATE(X)
IF(false, X, Y) -> ACTIVATE(Y)
F(X) -> IF(X, c, nf(ntrue))

Rules:

f(X) -> if(X, c, nf(ntrue))
f(X) -> nf(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> ntrue
activate(nf(X)) -> f(activate(X))
activate(ntrue) -> true
activate(X) -> X

Strategy:

innermost

Using the Dependency Graph the DP problem was split into 1 DP problems.

R
DPs
→DP Problem 1
AFS
→DP Problem 2
DGraph
...
→DP Problem 3
Argument Filtering and Ordering

Dependency Pair:

ACTIVATE(nf(X)) -> ACTIVATE(X)

Rules:

f(X) -> if(X, c, nf(ntrue))
f(X) -> nf(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> ntrue
activate(nf(X)) -> f(activate(X))
activate(ntrue) -> true
activate(X) -> X

Strategy:

innermost

The following dependency pair can be strictly oriented:

ACTIVATE(nf(X)) -> ACTIVATE(X)

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(n__f(x1)) =  1 + x1 POL(ACTIVATE(x1)) =  x1

resulting in one new DP problem.
Used Argument Filtering System:
ACTIVATE(x1) -> ACTIVATE(x1)
nf(x1) -> nf(x1)

R
DPs
→DP Problem 1
AFS
→DP Problem 2
DGraph
...
→DP Problem 4
Dependency Graph

Dependency Pair:

Rules:

f(X) -> if(X, c, nf(ntrue))
f(X) -> nf(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> ntrue
activate(nf(X)) -> f(activate(X))
activate(ntrue) -> true
activate(X) -> X

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:00 minutes