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(.module:
[lux #+ ->]
(lux (control [monad #+ do]
["p" parser "parser/" Monad<Parser>]
["ex" exception #+ exception:])
(data [product]
["e" error]
text/format
(coll [list "list/" Functor<List>]))
(lang [type])
(type abstract)
[macro]
(macro [code]
["s" syntax #+ syntax:]
(syntax (common ["scr" reader]
["scw" writer]))
[poly])))
(abstract: #export (Witness t w)
{}
t
(.def: #export !
(.All [t]
(.Ex [w]
(-> t (Witness t w))))
(.|>> @abstraction))
(.def: #export ?
(.All [t w]
(-> (Witness t w) t))
(.|>> @representation))
)
(syntax: #export (@ [name s.symbol])
(do @
[witnessT (macro.find-type name)]
(.case (.<| (poly.run witnessT)
poly.apply
(p.after (poly.this Witness))
(p.after poly.any)
poly.any)
(#e.Success identityT)
(wrap (.list (type.to-code identityT)))
(#e.Error error)
(macro.fail error))))
(syntax: #export (proof: [export? scr.export]
[name s.local-symbol]
type
proof)
(do @
[g!w (macro.gensym "w")]
(wrap (.list (.` (.def: (~+ (scw.export export?))
(~ (code.local-symbol name))
(..! (.: (~ type)
(~ proof)))))))))
(.def: type-vars
(s.Syntax (.List .Text))
(p.default (.list)
(s.tuple (p.some s.local-symbol))))
(.type: Input'
{#input-name .Text
#input-type .Code})
(.type: Input
(#Identified Input')
(#Anonymous Input'))
(.do-template [<name> <tag> <type>]
[(.def: (<name> input)
(-> Input <type>)
(.case input
(#Identified input')
(.get@ <tag> input')
(#Anonymous input')
(.get@ <tag> input')))]
[input-name #input-name .Text]
[input-type #input-type .Code]
)
(.def: (identified? input)
(-> Input .Bool)
(.case input
(#Identified input')
true
(#Anonymous input')
false))
(.def: input
(s.Syntax Input)
(p.alt (s.record (p.seq s.local-symbol s.any))
(s.tuple (p.seq s.local-symbol s.any))))
(.def: theorem-declaration
(s.Syntax [.Text (.List Input)])
(s.form (p.seq s.local-symbol (p.some input))))
(.def: proposition-declaration
(s.Syntax [.Text (.List .Text)])
(s.form (p.seq s.local-symbol (p.many s.local-symbol))))
(syntax: #export (proposition: [export? scr.export]
[[name elements] proposition-declaration]
[meaning (p.maybe s.any)])
(.case meaning
#.None
(wrap (.list (.` ((~! abstract:) (~+ (scw.export export?))
((~ (code.local-symbol name))
(~+ (list/map code.local-symbol elements)))
{}
.Unit))))
(#.Some meaning)
(wrap (.list (.` (.type: (~+ (scw.export export?))
((~ (code.local-symbol name))
(~+ (list/map code.local-symbol elements)))
(~ meaning)))))))
(.def: (theorem-type type-vars inputs meaning)
(-> (.List .Text) (.List Input) .Code .Code)
(.let [g!universals (list/map code.local-symbol
type-vars)
g!identities (.|> inputs
(list.filter identified?)
(list/map (.|>> input-name code.local-symbol)))
g!requisites (list/map (.function [input]
(.case input
(#Identified input')
(.` (..Witness (~ (.get@ #input-type input'))
(~ (code.local-symbol (input-name input)))))
(#Anonymous input')
(.get@ #input-type input')))
inputs)]
(.` (.All [(~+ g!universals)]
(.Ex [(~+ g!identities)]
(-> (~+ g!requisites) (~ meaning)))))))
(syntax: #export (axiom [description (p.default "" s.text)])
(wrap (.list (.` (.:!! [])))))
(syntax: #export (theorem [type-vars type-vars]
[[name inputs] theorem-declaration]
outputT
meaning)
(.let [inputs-names (list/map (.|>> input-name code.local-symbol)
inputs)]
(wrap (.list (.` (.: (~ (theorem-type type-vars inputs outputT))
(.function (~ (code.local-symbol name))
[(~+ inputs-names)]
(~ meaning))))))))
(.def: (input-code input)
(-> Input .Code)
(.case input
(#Identified [name type])
(.` {(~ (code.local-symbol name)) (~ type)})
(#Anonymous [name type])
(.` [(~ (code.local-symbol name)) (~ type)])))
(syntax: #export (theorem: [export? scr.export]
[type-vars type-vars]
[[name inputs] theorem-declaration]
outputT
meaning)
(wrap (.list (.` (..proof: (~+ (scw.export export?))
(~ (code.local-symbol name))
(~ (theorem-type type-vars inputs outputT))
(..theorem [(~+ (list/map code.local-symbol type-vars))]
((~ (code.local-symbol name)) (~+ (list/map input-code inputs)))
(~ outputT)
(~ meaning)))))))
(.type: #export (not p)
(-> p .Bottom))
(.type: #export (Test p)
(#True p)
(#False (not p)))
(exception: #export Absurdity)
(.def: #export absurdity
(.All [p] (-> p .Bottom))
(.function [proof]
(.error! (ex.construct Absurdity []))))
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