Termination Proof

Input TRS

Termination of the rewrite relation of the following TRS is considered.

g(c,g(c,x)) g(e,g(d,x))
g(d,g(d,x)) g(c,g(e,x))
g(e,g(e,x)) g(d,g(c,x))
f(g(x,y)) g(y,g(f(f(x)),a))
g(x,g(y,g(x,y))) g(a,g(x,g(y,b)))

Proof

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

g#(c,g(c,x)) g#(d,x)
g#(c,g(c,x)) g#(e,g(d,x))
g#(d,g(d,x)) g#(e,x)
g#(d,g(d,x)) g#(c,g(e,x))
g#(e,g(e,x)) g#(c,x)
g#(e,g(e,x)) g#(d,g(c,x))
f#(g(x,y)) f#(x)
f#(g(x,y)) f#(f(x))
f#(g(x,y)) g#(f(f(x)),a)
f#(g(x,y)) g#(y,g(f(f(x)),a))
g#(x,g(y,g(x,y))) g#(y,b)
g#(x,g(y,g(x,y))) g#(x,g(y,b))
g#(x,g(y,g(x,y))) g#(a,g(x,g(y,b)))

1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.