import Data.Set (Set)
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-- Slides
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data Term =
    Nat Integer
    | Abs String Term   -- EAbs Abstraction
    | Inv Term Term     -- EApp Invocation (a.k.a. Application) of a function
    | Var String        -- EVar a variable
    | Add

pretty x = case x of
    (Abs param term) -> "L" ++ param ++ "." ++ pretty term
    (Inv func param) -> "(" ++ pretty func ++ " " ++ pretty param ++ ")"
    (Var name) -> name
    Add -> "Add"
    Nat value -> show value

-- example
-- >>> pretty (Abs "x" $ Abs "y" $ Inv (Inv Add (Var "x")) (Var "y"))
-- "Lx.Ly.((Add x) y)"

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-- Exercise 1
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-- Ja
-- Nein
-- Ja
-- Nein

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-- Exercise 2
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-- Rule 1: Function bodies in (L{param}.{body}) span as much as possible:
--              Lx.A B C == Lx.(A B C)
-- Rule 2: Function invocations (application) are left bound
--              A B C == ((A B) C)

-- ((Lx.(x z)) (Ly.(x y)))
-- ((Lx.(x z)) (Ly.(w (Lw.(((w y) z) x)))))
-- (Lx.((x y) (Lx.(y x))))

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-- Exercise 3
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-- (Lx.((x *z*) (Ly.(x y))))
-- ((Lx.(x *z*)) Ly.(*w* (Lw.(((w y) *z*) *x*))))
-- Lx.((x "y") (Lx.("y" x)))

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-- Exercise 4
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----------
-- 1.)
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-- (((Lz.z) (Ly.yy)) (Lx.xa))
-- ((Ly.yy) (Lx.xa))
-- (Lx.xa (Lx.xa))
-- (Lx.xa) a
-- (a a)

----------
-- 2.)
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-- (Lx.(x x))(Ly.(y x)) z
-- (Ly.(y x)) (Ly.(y x)) z
-- ((Ly.(y x)) x) z
-- (x x) z
-- x x z

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-- 3.)
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-- ((Lx.(x x))(Ly.y))(Ly.y)
-- ((Ly.y Ly.y))(Ly.y)
-- (Ly.y)(Ly.y)
-- (Ly.y)