(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
.(1, x) → x
.(x, 1) → x
.(i(x), x) → 1
.(x, i(x)) → 1
i(1) → 1
i(i(x)) → x
.(i(y), .(y, z)) → z
.(y, .(i(y), z)) → z
.(.(x, y), z) → .(x, .(y, z))
i(.(x, y)) → .(i(y), i(x))
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(.(x1, x2)) = 2 + x1 + x2
POL(1) = 1
POL(i(x1)) = 2·x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
.(1, x) → x
.(x, 1) → x
.(i(x), x) → 1
.(x, i(x)) → 1
i(1) → 1
.(i(y), .(y, z)) → z
.(y, .(i(y), z)) → z
i(.(x, y)) → .(i(y), i(x))
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
i(i(x)) → x
.(.(x, y), z) → .(x, .(y, z))
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(.(x1, x2)) = x1 + x2
POL(i(x1)) = 1 + x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
i(i(x)) → x
(4) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
.(.(x, y), z) → .(x, .(y, z))
Q is empty.
(5) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(.(x1, x2)) = 2 + 2·x1 + x2
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
.(.(x, y), z) → .(x, .(y, z))
(6) Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.
(7) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(8) TRUE
(9) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(10) TRUE
(11) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(12) TRUE