(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
if(true, x, y) → x
if(false, x, y) → y
if(x, y, y) → y
if(if(x, y, z), u, v) → if(x, if(y, u, v), if(z, u, v))
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(false) = 0
POL(if(x1, x2, x3)) = 2·x1 + 2·x2 + 2·x3
POL(true) = 2
POL(u) = 0
POL(v) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
if(true, x, y) → x
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
if(false, x, y) → y
if(x, y, y) → y
if(if(x, y, z), u, v) → if(x, if(y, u, v), if(z, u, v))
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(false) = 2
POL(if(x1, x2, x3)) = 2·x1 + 2·x2 + 2·x3
POL(u) = 0
POL(v) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
if(false, x, y) → y
(4) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
if(x, y, y) → y
if(if(x, y, z), u, v) → if(x, if(y, u, v), if(z, u, v))
Q is empty.
(5) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(if(x1, x2, x3)) = 1 + 2·x1 + x2 + x3
POL(u) = 0
POL(v) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
if(x, y, y) → y
(6) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
if(if(x, y, z), u, v) → if(x, if(y, u, v), if(z, u, v))
Q is empty.
(7) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Recursive path order with status [RPO].
Quasi-Precedence:
[u, v] > if3
Status:
if3: [1,3,2]
u: multiset
v: multiset
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
if(if(x, y, z), u, v) → if(x, if(y, u, v), if(z, u, v))
(8) Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.
(9) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(10) YES