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% An Implementation of G4ip for Terzo

% 	author: Christian.Urban@cl.cam.ac.uk 

% 

% some solvable sample queries:

%

% prove (nil |- p imp p).

% prove (nil |- (p imp p) imp (p imp p)).

% prove (nil |- ((((p imp q) imp p) imp p) imp q) imp q).

% prove (nil |- (a imp (b imp c)) imp ((a imp b) imp (a imp c))).

% prove (nil |- (a or (a imp b)) imp (((a imp b) imp a) imp a)).

%

% two non-solvable queries

%

% prove (nil |- a or (a imp false)).

% prove (nil |- ((a imp b) imp a) imp a).

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module G4ip.

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% atomic formulae

kind form type.

type p form. 

type q form.

type a form.

type b form.

type c form.

type isatomic form -> o.

isatomic p.

isatomic q.

isatomic a.

isatomic b.

isatomic c.

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% logical operators

type false  form.

type and    form -> form -> form.    

type or     form -> form -> form.       

type imp    form -> form -> form.

infixr and   9.

infixr or    9.

infixr imp   9.

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% sequent constructor; sequents are of the form: (list |- formula) 

kind seq type.

type   |-  list form -> form -> seq. 

infixl |-   4.

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% prove predicate; prints "solvable" if seq is provable, otherwise "no" 

type prove seq -> o.

prove (Gamma |- G) :- ( membNrest P Gamma Gamma',

                        left (P::Gamma' |- G) );

                        right   (Gamma  |- G).   

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% rightrules

type right seq -> o.

right (Gamma |- B and C) :- prove (Gamma |- B),          %% and-R   

                            prove (Gamma |- C).          

right (Gamma |- B imp C) :- prove (B::Gamma |- C).       %% imp-R

right (Gamma |- B or  C) :- prove (Gamma |- B);          %% or-R 

                            prove (Gamma |- C).

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% leftrules

type left  seq -> o.

left (false :: Gamma |- G).                              %% false-L

left (A :: Gamma |- A)     :- isatomic A.                %% axiom 

left (B and C::Gamma |- G) :- prove (B::C::Gamma |- G).  %% and-L

left (B  or C::Gamma |- G) :- prove (B::Gamma |- G),     %% or-L 

                              prove (C::Gamma |- G).

left (A imp B::Gamma |- G) :-                            %% imp-L1

	isatomic A, ismember A Gamma, prove (B::Gamma |- G).

left ((B and C) imp D::Gamma |- G) :-                    %% imp-L2

	prove (B imp (C imp D)::Gamma |- G).

left ((B or  C) imp D::Gamma |- G) :-                    %% imp-L3

	prove (B imp D::C imp D::Gamma |- G).

left ((B imp C) imp D::Gamma |- G) :-                    %% imp-L4

	prove (C imp D::Gamma |- B imp C),	                              

	prove (D::Gamma |- G).

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% returns a member and the remainder of a list

type membNrest     A -> list A -> list A -> o.

membNrest X (X::Rest) Rest.

membNrest X (Y::Tail) (Y::Rest) :- membNrest X Tail Rest.

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% succeeds only once if A is element in the list 

type ismember     A -> list A -> o.

ismember X (X::Rest) :- !.

ismember X (Y::Tail) :- ismember X Tail.