%%% Constructive Logic (15-317), Fall 2009
%%% Assignment 9: Proving Metatheorems in Twelf

%% Atoms, including some sample ones for examples
atom : type.  %name atom P.
a : atom.
b : atom.
c : atom.
d : atom.
e : atom.
f : atom.

%% Propositions
prop : type.  %name prop A.
\/ : prop -> prop -> prop.  %infix right 12 \/.
~  : prop -> prop.          %prefix 13 ~.
?  : atom -> prop.          %prefix 14 ?.

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%%  TODO (Task 2): define =>  %%
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% (uncomment to get fixity, once you've defined => above. -wjl)
% %infix right 10 =>.


%% Classical Sequent Calculus

contra : type.
true : prop -> type.
false : prop -> type.

init : true (? P) -> false (? P) -> contra.

\/R : (false A -> false B -> contra)
   -> (false (A \/ B) -> contra).

\/L : (true A -> contra)
   -> (true B -> contra)
   -> (true (A \/ B) -> contra).

~R : (true A -> contra)
  -> (false (~ A) -> contra).

~L : (false A -> contra)
  -> (true (~ A) -> contra).

%% Derived rules

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%%  TODO (Task 2): define =>R, =>L  %%
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%% Identity Theorem

identity : {A:prop} (true A -> false A -> contra) -> type.
%mode identity +A -D.

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%%  TODO (Task 3): fill in cases  %%
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%worlds () (identity _ _).
%total A (identity A _).