Created
May 15, 2020 16:31
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(A subset of) the two-sided linear logic sequent calculus
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delete :: (Eq a) => a -> [a] -> [a] | |
delete _ [] = error "delete: element not present" | |
delete x (y:ys) = if x == y then ys else y : delete x ys | |
data Atom | |
= Var String | |
| Constant String | |
deriving (Eq,Ord,Show) | |
data Exp | |
= Atom Atom | |
| Perp Exp | |
| Bot | |
| Zero | |
| Plus Exp Exp | |
| Lolli Exp Exp | |
| Bang Exp | |
| Quest Exp | |
deriving (Eq,Ord,Show) | |
dual :: Exp -> Exp | |
dual a@(Atom _) = Perp a | |
dual (Perp a) = a | |
dual (Bang e) = Quest (dual e) | |
dual (Quest e) = Bang (dual e) | |
dual (Lolli x Bot) = x | |
dual (Lolli x y) = Lolli (dual y) (dual x) | |
dual x = Perp x | |
async Bot = True | |
async (Lolli _ _) = True | |
async (Quest _) = True | |
async (Perp a) = not (async a) | |
async _ = False | |
data Seq = Seq [Exp] [Exp] | |
deriving (Eq,Ord,Show) | |
par a b = Lolli (Perp a) b | |
tensor a b = Perp (Lolli a (Perp b)) | |
with a b = Perp (Plus (Perp a) (Perp b)) | |
unit = Perp Bot | |
top = Perp Zero | |
data Rule | |
= Identity Exp | |
| Cut Exp Seq Seq | |
| PerpL Exp Seq | |
| PerpR Exp Seq | |
| PlusL Exp Exp Seq Seq | |
| PlusR Exp Exp Seq Seq | |
| BotR Seq | |
| BotL | |
| ZeroL Seq | |
| ImplR Exp Exp Seq | |
| ImplL Exp Exp Seq Seq | |
| WeakenL Exp Seq | |
| WeakenR Exp Seq | |
| ContractL Exp Seq | |
| ContractR Exp Seq | |
| BangL Exp Seq | |
| QuestR Exp Seq | |
| BangR Exp Seq | |
| QuestL Exp Seq | |
deriving (Eq,Ord,Show) | |
rule (Identity a) = Seq [a] [a] | |
rule (Cut a (Seq w x) (Seq y z)) = Seq (w ++ delete a y) (delete a x ++ z) | |
rule (PerpL a (Seq x y)) = Seq (x ++ [Perp a]) (delete a y) | |
rule (PerpR a (Seq x y)) = Seq (delete a x) ([Perp a] ++ y) | |
rule (PlusL a b (Seq w x) (Seq y z)) | |
| delete a w == delete b y && x == z | |
= Seq (delete a w ++ [Plus a b]) z | |
rule (PlusR a b (Seq w x) (Seq y z)) | |
| delete a x == delete b z && w == y | |
= Seq w (delete a x ++ [Plus a b]) | |
rule (BotR (Seq x y)) = Seq x (y ++ [Bot]) | |
rule BotL = Seq [Bot] [] | |
rule (ZeroL (Seq x y)) = Seq (x ++ [Zero]) y | |
rule (ImplR a b (Seq x y)) | |
= Seq (delete a x) (delete b y ++ [Lolli a b]) | |
rule (ImplL a b (Seq w x) (Seq y z)) | |
= Seq ([Lolli a b] ++ w ++ delete b y) (delete a x ++ z) | |
rule (WeakenL a (Seq x y)) = Seq (x ++ [Bang a]) y | |
rule (WeakenR a (Seq x y)) = Seq x (y ++ [Quest a]) | |
rule (ContractL a (Seq x y)) = | |
let aa = Bang a | |
xx = [aa] ++ delete aa (delete aa x) | |
in Seq xx y | |
rule (ContractR a (Seq x y)) = | |
let aa = Quest a | |
yy = [aa] ++ delete aa (delete aa y) | |
in Seq x yy | |
rule (BangL a (Seq x y)) = Seq ([Bang a] ++ delete a x) y | |
rule (QuestR a (Seq x y)) = Seq x ([Quest a] ++ delete a y) | |
rule (BangR a (Seq x y)) = Seq x ([Bang a] ++ delete a y) -- assert: x is all Bang's, delete a y is all Quest's | |
rule (QuestL a (Seq x y)) = Seq ([Quest a] ++ delete a x) y -- assert: delete a x is all Bang's, y is all Quest's |
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