%Copyright 2020 Marianna Girlando, Lutz Straßburger.
%
%This file is part of MOIN. MOIN is free software: you can redistribute it
%and or modify it under the terms of the GNU General Public License as published
%by the Free Software Foundation, version 3 of the License.
%
%MOIN is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY;
%without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR
%PURPOSE. See the GNU General Public License for more details. You should have
%received a copy of the GNU General Public License along with MOIN. If not, see
%<http://www.gnu.org/licenses/>.


% Axioms of S5-cube
k1. (!(a->b)) -> ((!a)->(!b))
d. !a -> ?a
t1. a -> ?a
t2. !a -> a
b1. a -> ! (?a)
b2. ? (!a)-> a
4.1. !a -> !(!a)
4.2. ?(?a)-> ?a
5.1. ?a -> !(?a)
5.2. ?(!a) -> !a

%Axioms of IS5-cube
k1. (!(a->b)) -> ((!a)->(!b))
k2. (!(a->b)) -> ((?a)->(?b))
k3. (?(a v b)) -> ((?a)v(?b))
k4. ((?a)-> (!b)) -> (!(a->b))
k5. (?false)->false
d. !a-> ?a
t. (a->(?a))^((!a)->a)
b. (a->(!(?a)))^(((?(!a)))->a)
4. ((?(?a))->(?a))^((!a)->(!(!a)))
5. (?a->(!(?a))) ^ ( (?(!a))->!a )


% run from bash

./moin.sh KT4 '(?(a v b)) -> ((?a)v(?b))'

./moin.sh S4 '?(!a) -> !a'
./moin.sh S5 '?(!a) -> !a'

./moin.sh IS4s '(?(a v b)) -> ((?a)v(?b))'

./moin.sh IK4m '(?(a v b)) -> ((?a)v(?b))'

./moin.sh IK5s '? (!a)-> a'
./moin.sh IK5m '? (!a)-> a'
./moin.sh IS5s '? (!a)-> a'
./moin.sh IS5m '? (!a)-> a'
