(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) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(.(x1, x2)) = 2·x1 + 2·x2
POL(a) = 0
POL(b) = 0
POL(b') = 2
POL(c) = 0
POL(d) = 0
POL(d') = 0
POL(e) = 0
POL(f(x1, x2)) = 2·x1 + 2·x2
POL(g(x1, x2)) = 2·x1 + x2
POL(h(x1, x2)) = x1 + 2·x2
POL(i(x1, x2, x3)) = 1 + x1 + 2·x2 + 2·x3
POL(if(x1, x2, x3)) = 2·x1 + 2·x2 + 2·x3
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
f(g(i(a, b, b'), c), d) → if(e, f(.(b, c), d'), f(.(b', c), d'))
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(g(h(a, b), c), d) → if(e, f(.(b, g(h(a, b), c)), d), f(c, d'))
Q is empty.
(3) Overlay + Local Confluence (EQUIVALENT transformation)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
(4) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
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(h(a, b), c), d)
(5) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
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(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(h(a, b), c), d)
We have to consider all minimal (P,Q,R)-chains.
(7) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
(8) TRUE