(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(g(i(a, b, b'), c), d) → if(e, f(.(b, c), d'), f(.(b', c), d'))
f(g(h(a, b), c), d) → if(e, f(.(b, g(h(a, b), c)), d), f(c, d'))
Q is empty.
(1) Overlay + Local Confluence (EQUIVALENT transformation)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(g(i(a, b, b'), c), d) → if(e, f(.(b, c), d'), f(.(b', c), d'))
f(g(h(a, b), c), d) → if(e, f(.(b, g(h(a, b), c)), d), f(c, d'))
The set Q consists of the following terms:
f(g(i(a, b, b'), c), d)
f(g(h(a, b), c), d)
(3) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(g(i(a, b, b'), c), d) → F(.(b, c), d')
F(g(i(a, b, b'), c), d) → F(.(b', c), d')
F(g(h(a, b), c), d) → F(.(b, g(h(a, b), c)), d)
F(g(h(a, b), c), d) → F(c, d')
The TRS R consists of the following rules:
f(g(i(a, b, b'), c), d) → if(e, f(.(b, c), d'), f(.(b', c), d'))
f(g(h(a, b), c), d) → if(e, f(.(b, g(h(a, b), c)), d), f(c, d'))
The set Q consists of the following terms:
f(g(i(a, b, b'), c), d)
f(g(h(a, b), c), d)
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 4 less nodes.
(6) TRUE