; @origtpdbfilename ./TRS/secret06/aprove/division.trs
; @xtcfilename "./TRS_Standard/Secret_06_TRS/division.xml"
(format TRS)
(fun division 2)
(fun div 3)
(fun |0| 0)
(fun if 4)
(fun lt 2)
(fun inc 1)
(fun true 0)
(fun false 0)
(fun s 1)
(fun minus 2)
(rule (division x y) (div x y |0|))
(rule (div x y z) (if (lt x y) x y (inc z)))
(rule (if true x y z) z)
(rule (if false x (s y) z) (div (minus x (s y)) (s y) z))
(rule (minus x |0|) x)
(rule (minus (s x) (s y)) (minus x y))
(rule (lt x |0|) false)
(rule (lt |0| (s y)) true)
(rule (lt (s x) (s y)) (lt x y))
(rule (inc |0|) (s |0|))
(rule (inc (s x)) (s (inc x)))