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