NO

The TRS could be proven non-terminating. The proof took 223 ms.

The following reduction sequence is a witness for non-termination:

g#(___x) →* g#(___x)

The following DP Processors were used


Problem 1 was processed with processor DependencyGraph (6ms).
 | – Problem 2 was processed with processor BackwardInstantiation (1ms).
 |    | – Problem 3 was processed with processor BackwardInstantiation (1ms).
 |    |    | – Problem 4 was processed with processor BackwardInstantiation (1ms).
 |    |    |    | – Problem 5 remains open; application of the following processors failed [ForwardInstantiation (2ms), Propagation (0ms)].

Problem 1: DependencyGraph



Dependency Pair Problem

Dependency Pairs

g#(x)g#(x)g#(x)f#(g(x))

Rewrite Rules

f(g(a))af(f(x))b
g(x)f(g(x))

Original Signature

Termination of terms over the following signature is verified: f, g, b, a

Strategy


The following SCCs where found

g#(x) → g#(x)

Problem 2: BackwardInstantiation



Dependency Pair Problem

Dependency Pairs

g#(x)g#(x)

Rewrite Rules

f(g(a))af(f(x))b
g(x)f(g(x))

Original Signature

Termination of terms over the following signature is verified: f, g, b, a

Strategy


Instantiation

For all potential predecessors l → r of the rule g#(x) → g#(x) on dependency pair chains it holds that: Thus, g#(x) → g#(x) is replaced by instances determined through the above matching. These instances are:
g#(_x) → g#(_x)

Problem 3: BackwardInstantiation



Dependency Pair Problem

Dependency Pairs

g#(_x)g#(_x)

Rewrite Rules

f(g(a))af(f(x))b
g(x)f(g(x))

Original Signature

Termination of terms over the following signature is verified: f, g, b, a

Strategy


Instantiation

For all potential predecessors l → r of the rule g#(_x) → g#(_x) on dependency pair chains it holds that: Thus, g#(_x) → g#(_x) is replaced by instances determined through the above matching. These instances are:
g#(__x) → g#(__x)

Problem 4: BackwardInstantiation



Dependency Pair Problem

Dependency Pairs

g#(__x)g#(__x)

Rewrite Rules

f(g(a))af(f(x))b
g(x)f(g(x))

Original Signature

Termination of terms over the following signature is verified: f, g, b, a

Strategy


Instantiation

For all potential predecessors l → r of the rule g#(__x) → g#(__x) on dependency pair chains it holds that: Thus, g#(__x) → g#(__x) is replaced by instances determined through the above matching. These instances are:
g#(___x) → g#(___x)