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(* Title: FOLP/FOLP_lemmas.ML

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ID: $Id$


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Author: Martin D Coen, Cambridge University Computer Laboratory


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Copyright 1991 University of Cambridge


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*)


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(*** Classical introduction rules for  and EX ***)


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val prems= goal (the_context ())


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"(!!x. x:~Q ==> f(x):P) ==> ?p : PQ";


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by (rtac classical 1);


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by (REPEAT (ares_tac (prems@[disjI1,notI]) 1));


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by (REPEAT (ares_tac (prems@[disjI2,notE]) 1)) ;


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qed "disjCI";


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(*introduction rule involving only EX*)


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val prems= goal (the_context ())


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"( !!u. u:~(EX x. P(x)) ==> f(u):P(a)) ==> ?p : EX x. P(x)";


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by (rtac classical 1);


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by (eresolve_tac (prems RL [exI]) 1) ;


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qed "ex_classical";


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(*version of above, simplifying ~EX to ALL~ *)


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val [prem]= goal (the_context ())


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"(!!u. u:ALL x. ~P(x) ==> f(u):P(a)) ==> ?p : EX x. P(x)";


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by (rtac ex_classical 1);


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by (resolve_tac [notI RS allI RS prem] 1);


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by (etac notE 1);


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by (etac exI 1) ;


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qed "exCI";


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val excluded_middle = prove_goal (the_context ()) "?p : ~P  P"


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(fn _=> [ rtac disjCI 1, assume_tac 1 ]);


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(*** Special elimination rules *)


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(*Classical implies (>) elimination. *)


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val major::prems= goal (the_context ())


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"[ p:P>Q; !!x. x:~P ==> f(x):R; !!y. y:Q ==> g(y):R ] ==> ?p : R";


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by (resolve_tac [excluded_middle RS disjE] 1);


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by (DEPTH_SOLVE (ares_tac (prems@[major RS mp]) 1)) ;


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qed "impCE";


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(*Double negation law*)


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Goal "p:~~P ==> ?p : P";


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by (rtac classical 1);


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by (etac notE 1);


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by (assume_tac 1);


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qed "notnotD";


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(*** Tactics for implication and contradiction ***)


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(*Classical <> elimination. Proof substitutes P=Q in


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~P ==> ~Q and P ==> Q *)


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val prems = goalw (the_context ()) [iff_def]


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"[ p:P<>Q; !!x y.[ x:P; y:Q ] ==> f(x,y):R; \


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\ !!x y.[ x:~P; y:~Q ] ==> g(x,y):R ] ==> ?p : R";


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by (rtac conjE 1);


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by (REPEAT (DEPTH_SOLVE_1 (etac impCE 1


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ORELSE mp_tac 1 ORELSE ares_tac prems 1))) ;


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qed "iffCE";


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(*Should be used as swap since ~P becomes redundant*)


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val major::prems= goal (the_context ())


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"p:~P ==> (!!x. x:~Q ==> f(x):P) ==> ?p : Q";


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by (rtac classical 1);


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by (rtac (major RS notE) 1);


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by (REPEAT (ares_tac prems 1)) ;


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qed "swap";
