; @origtpdbfilename ./TRS/Thiemann/div_notTermin.trs
; @xtcfilename "./TRS_Standard/AProVE_06/div_notTermin.xml"
(format TRS)
(fun ge 2)
(fun |0| 0)
(fun true 0)
(fun s 1)
(fun false 0)
(fun minus 2)
(fun plus 2)
(fun div 2)
(fun ify 3)
(fun divByZeroError 0)
(fun if 3)
(rule (ge |0| |0|) true)
(rule (ge (s x) |0|) (ge x |0|))
(rule (ge |0| (s |0|)) false)
(rule (ge |0| (s (s x))) (ge |0| (s x)))
(rule (ge (s x) (s y)) (ge x y))
(rule (minus |0| |0|) |0|)
(rule (minus |0| (s x)) (minus |0| x))
(rule (minus (s x) |0|) (s (minus x |0|)))
(rule (minus (s x) (s y)) (minus x y))
(rule (plus |0| |0|) |0|)
(rule (plus |0| (s x)) (s (plus |0| x)))
(rule (plus (s x) y) (s (plus x y)))
(rule (div x y) (ify (ge y (s |0|)) x y))
(rule (ify false x y) divByZeroError)
(rule (ify true x y) (if (ge x y) x y))
(rule (if false x y) |0|)
(rule (if true x y) (s (div (minus x y) y)))
(rule (div (plus x y) z) (plus (div x z) (div y z)))