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module Inst (Gnd : Set)(U : Set)(El : U -> Set) where
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winst : {G : Cxt}{C : Kind}{L : Loc}{I : Kind}
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(x : L ! I) -> G [ L bar x / Term C ]- I ->
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{T : Kind}(t : L [ Term C ]- T) -> [| Good G t |] ->
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G [ L bar x / Term C ]- T
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winst x i [ s ] sg = G[ winsts x i s sg ]
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winst x i (fn A f) fg = Gfn A \ a -> winst x i (f a) (fg a)
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winst x i (\\ b) bg = G\\ (winst (pop x) (shift popH i) b bg)
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winst x i (h $ s) pg = wing x i h (fst pg) (winsts x i s (snd pg))
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winsts : {G : Cxt}{C : Kind}{L : Loc}{Z : Gnd}{I : Kind}
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(x : L ! I) -> G [ L bar x / Term C ]- I ->
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{T : Kind}(s : L [ Args C Z ]- T) -> [| Good G s |] ->
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G [ L bar x / Args C Z ]- T
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winsts x i (a ^ s) sg = a G^ winsts x i s sg
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winsts x i (r & s) pg = winst x i r (fst pg) G& winsts x i s (snd pg)
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winsts x i nil _ = Gnil
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wing : {G : Cxt}{C : Kind}{L : Loc}{Z : Gnd}{I : Kind}
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(x : L ! I) -> G [ L bar x / Term C ]- I ->
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{T : Kind}(h : L [ Head ]- T) -> [| Good G h |] ->
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G [ L bar x / Args C Z ]- T ->
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G [ L bar x / Term C ]- Ty Z
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wing x i (` k) kg s = (` k -! kg) G$ s
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wing x i (# y) _ s with varQV x y
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wing x i (# .x) _ s | vSame = go i s
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wing x i (# .(x thin y)) _ s | vDiff y = (# y -! _) G$ s
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go : {G : Cxt}{C : Kind}{L : Loc}{Z : Gnd}{I : Kind} ->
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G [ L / Term C ]- I -> G [ L / Args C Z ]- I ->
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G [ L / Term C ]- Ty Z
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go (fn A f -! fg) ((a ^ s) -! sg) = go (f a -! fg a) (s -! sg)
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go (\\ b -! bg) ((r & s) -! pg) =
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go (winst top (r -! fst pg) b bg) (s -! snd pg)
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inst : {G : Cxt}{C : Kind}{L : Loc}{I : Kind}
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(x : L ! I) -> G [ L bar x / Term C ]- I ->
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{T : Kind} -> G [ L / Term C ]- T -> G [ L bar x / Term C ]- T
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inst x i (t -! tg) = winst x i t tg
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_$$_ : {G : Cxt}{C S T : Kind}{L : Loc} ->
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G [! C !]- (S |> T) -> G [! C !]- S -> G [! C !]- T
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(\\ b -! bg) $$ sg = inst top sg (b -! bg)