addWith.golden

MiniAgda by Andreas Abel and Karl Mehltretter
--- opening "addWith.ma" ---
--- scope checking ---
--- type checking ---
type  SNat : + Size -> Set
term  SNat.zero : .[s!ze : Size] -> .[i < s!ze] -> SNat s!ze
term  SNat.zero : .[i : Size] -> < SNat.zero i : SNat $i >
term  SNat.succ : .[s!ze : Size] -> .[i < s!ze] -> ^ SNat i -> SNat s!ze
term  SNat.succ : .[i : Size] -> ^(y1 : SNat i) -> < SNat.succ i y1 : SNat $i >
term  ote : .[i : Size] -> SNat i -> SNat i
{ ote [.$i] (SNat.zero [i]) = SNat.zero [i]
; ote [.$$i] (SNat.succ [.$i] (SNat.zero [i])) = SNat.zero [i]
; ote [.$$i] (SNat.succ [.$i] (SNat.succ [i] x)) = SNat.succ [$i] (SNat.succ [i] (ote [i] x))
}
term  addWith : (.[k : Size] -> SNat k -> SNat k) -> .[i : Size] -> .[j : Size] -> SNat i -> SNat j -> SNat #
{ addWith f [.$i] [j] (SNat.zero [i]) y = y
; addWith f [.$i] [j] (SNat.succ [i] x) y = SNat.succ [#] (addWith f [j] [i] (f [j] y) (f [i] x))
}
term  three : SNat #
term  three = SNat.succ [#] (SNat.succ [#] (SNat.succ [#] (SNat.zero [#])))
term  four : SNat #
term  four = SNat.succ [#] three
term  bla : SNat #
term  bla = addWith ote [#] [#] four three
--- evaluating ---
bla has whnf SNat.succ{i = #; y1 = SNat.succ{i = #; y1 = SNat.succ{i = #; y1 = SNat.zero{i = #}}}}
bla evaluates to SNat.succ # (SNat.succ # (SNat.succ # (SNat.zero #)))
--- closing "addWith.ma" ---