module Proof

import Data.Vect

import Signature
import Term
import Formula
import Context

%default total

public export
data Proof : Context sig g -> Formula sig n -> Type where
  Hyp     : {c : Context sig g} -> (i : Fin g) -> Proof c (snd $ index i c)
  EqRefl  : (t : Term sig k) -> Proof c (Equal t t)
  EqSym   : Proof c (Equal s t) -> Proof c (Equal t s)
  EqTrans : Proof c (Equal s t) -> Proof c (Equal t u) -> Proof c (Equal s u)
  EqE     : {phi : Formula sig (S n)} -> Proof c (Equal s t) -> Proof c (elimBinder s phi) -> Proof c (elimBinder t phi)
  BotE    : Proof c Bot -> Proof c phi
  Contra  : Proof (Imp phi Bot :: c) Bot -> Proof c phi
  ImpI    : Proof (phi :: c) psi -> Proof c (Imp phi psi)
  ImpE    : Proof c (Imp phi psi) -> Proof c phi -> Proof c psi
  ForallI : Proof c phi -> Proof c (Forall phi)
  ForallE : Proof c (Forall phi) -> Proof c (elimBinder t phi)
  Weaken  : Proof c1 phi -> Subset c1 c2 -> Proof c2 phi