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% An Implementation of G4ip for Terzo+
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% 	author: Christian.Urban@cl.cam.ac.uk +
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% +
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% some solvable sample queries:+
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%+
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% prove (nil |- p imp p).+
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% prove (nil |- (p imp p) imp (p imp p)).+
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% prove (nil |- ((((p imp q) imp p) imp p) imp q) imp q).+
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% prove (nil |- (a imp (b imp c)) imp ((a imp b) imp (a imp c))).+
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% prove (nil |- (a or (a imp b)) imp (((a imp b) imp a) imp a)).+
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%+
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% two non-solvable queries+
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%+
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% prove (nil |- a or (a imp false)).+
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% prove (nil |- ((a imp b) imp a) imp a).+
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module G4ip.+
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% atomic formulae+
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kind form type.+
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type p form. +
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type q form.+
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type a form.+
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type b form.+
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type c form.+
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type isatomic form -> o.+
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isatomic p.+
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isatomic q.+
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isatomic a.+
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isatomic b.+
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isatomic c.+
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% logical operators+
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type false  form.+
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type and    form -> form -> form.    +
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type or     form -> form -> form.       +
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type imp    form -> form -> form.+
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 +
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infixr and   9.+
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infixr or    9.+
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infixr imp   9.+
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% sequent constructor; sequents are of the form: (list |- formula) +
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kind seq type.+
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type   |-  list form -> form -> seq. +
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infixl |-   4.+
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% prove predicate; prints "solvable" if seq is provable, otherwise "no" +
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type prove seq -> o.+
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prove (Gamma |- G) :- ( membNrest P Gamma Gamma',+
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                        left (P::Gamma' |- G) );+
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                        right   (Gamma  |- G).   +
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% rightrules+
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type right seq -> o.+
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right (Gamma |- B and C) :- prove (Gamma |- B),          %% and-R   +
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                            prove (Gamma |- C).          +
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right (Gamma |- B imp C) :- prove (B::Gamma |- C).       %% imp-R+
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right (Gamma |- B or  C) :- prove (Gamma |- B);          %% or-R +
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                            prove (Gamma |- C).+
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% leftrules+
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type left  seq -> o.+
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left (false :: Gamma |- G).                              %% false-L+
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left (A :: Gamma |- A)     :- isatomic A.                %% axiom +
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left (B and C::Gamma |- G) :- prove (B::C::Gamma |- G).  %% and-L+
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left (B  or C::Gamma |- G) :- prove (B::Gamma |- G),     %% or-L +
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                              prove (C::Gamma |- G).+
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left (A imp B::Gamma |- G) :-                            %% imp-L1+
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	isatomic A, ismember A Gamma, prove (B::Gamma |- G).+
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left ((B and C) imp D::Gamma |- G) :-                    %% imp-L2+
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	prove (B imp (C imp D)::Gamma |- G).+
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left ((B or  C) imp D::Gamma |- G) :-                    %% imp-L3+
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	prove (B imp D::C imp D::Gamma |- G).+
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left ((B imp C) imp D::Gamma |- G) :-                    %% imp-L4+
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	prove (C imp D::Gamma |- B imp C),	                              +
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	prove (D::Gamma |- G).+
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% returns a member and the remainder of a list+
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type membNrest     A -> list A -> list A -> o.+
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membNrest X (X::Rest) Rest.+
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membNrest X (Y::Tail) (Y::Rest) :- membNrest X Tail Rest.+
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% succeeds only once if A is element in the list +
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type ismember     A -> list A -> o.+
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ismember X (X::Rest) :- !.+
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ismember X (Y::Tail) :- ismember X Tail.+
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