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ArrMp.agda
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ArrMp.agda
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-- Minimal implicational logic, vector-based de Bruijn approach, initial encoding
module Vi.ArrMp where
open import Lib using (Nat; suc; _+_; Fin; fin; Vec; _,_; proj; VMem; mem)
-- Types
infixr 0 _=>_
data Ty : Set where
UNIT : Ty
_=>_ : Ty -> Ty -> Ty
-- Context and truth judgement
Cx : Nat -> Set
Cx n = Vec Ty n
isTrue : forall {tn} -> Ty -> Fin tn -> Cx tn -> Set
isTrue a i tc = VMem a i tc
-- Terms
module ArrMp where
infixr 0 lam=>_
infixl 1 _$_
data Tm {tn} (tc : Cx tn) : Ty -> Set where
var : forall {a i} -> isTrue a i tc -> Tm tc a
lam=>_ : forall {a b} -> Tm (tc , a) b -> Tm tc (a => b)
_$_ : forall {a b} -> Tm tc (a => b) -> Tm tc a -> Tm tc b
v : forall {tn} (k : Nat) {tc : Cx (suc (k + tn))} -> Tm tc (proj tc (fin k))
v i = var (mem i)
Thm : Ty -> Set
Thm a = forall {tn} {tc : Cx tn} -> Tm tc a
open ArrMp public
-- Example theorems
aI : forall {a} -> Thm (a => a)
aI =
lam=> v 0
aK : forall {a b} -> Thm (a => b => a)
aK =
lam=>
lam=> v 1
aS : forall {a b c} -> Thm ((a => b => c) => (a => b) => a => c)
aS =
lam=>
lam=>
lam=> v 2 $ v 0 $ (v 1 $ v 0)
tSKK : forall {a} -> Thm (a => a)
tSKK {a = a} =
aS {b = a => a} $ aK $ aK