author  clasohm 
Fri, 13 Oct 1995 11:48:40 +0100  
changeset 1280  909079af97b7 
parent 12  f17d542276b6 
permissions  rwrr 
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(* Title: FOL/ifol.ML 
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ID: $Id$ 

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Author: Lawrence C Paulson, Cambridge University Computer Laboratory 

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Copyright 1992 University of Cambridge 

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Tactics and lemmas for ifol.thy (intuitionistic firstorder logic) 

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*) 

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open IFOL; 

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signature IFOL_LEMMAS = 

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sig 

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val allE: thm 

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val all_cong: thm 

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val all_dupE: thm 

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val all_impE: thm 

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val box_equals: thm 

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val conjE: thm 

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val conj_cong: thm 

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val conj_impE: thm 

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val contrapos: thm 

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val disj_cong: thm 

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val disj_impE: thm 

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val eq_cong: thm 

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val eq_mp_tac: int > tactic 

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val ex1I: thm 

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val ex_ex1I: thm 
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val ex1E: thm 
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val ex1_equalsE: thm 

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val ex1_cong: thm 

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val ex_cong: thm 

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val ex_impE: thm 

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val iffD1: thm 

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val iffD2: thm 

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val iffE: thm 

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val iffI: thm 

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val iff_cong: thm 

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val iff_impE: thm 

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val iff_refl: thm 

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val iff_sym: thm 

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val iff_trans: thm 

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val impE: thm 

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val imp_cong: thm 

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val imp_impE: thm 

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val mp_tac: int > tactic 

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val notE: thm 

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val notI: thm 

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val not_cong: thm 

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val not_impE: thm 

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val not_sym: thm 

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val not_to_imp: thm 

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val pred1_cong: thm 

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val pred2_cong: thm 

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val pred3_cong: thm 

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val pred_congs: thm list 

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val rev_mp: thm 

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val simp_equals: thm 

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val ssubst: thm 

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val subst_context: thm 

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val subst_context2: thm 

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val subst_context3: thm 

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val sym: thm 

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val trans: thm 

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val TrueI: thm 

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end; 

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structure IFOL_Lemmas : IFOL_LEMMAS = 

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struct 

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val TrueI = prove_goalw IFOL.thy [True_def] "True" 

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(fn _ => [ (REPEAT (ares_tac [impI] 1)) ]); 

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(*** Sequentstyle elimination rules for & > and ALL ***) 

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val conjE = prove_goal IFOL.thy 

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"[ P&Q; [ P; Q ] ==> R ] ==> R" 

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(fn prems=> 

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[ (REPEAT (resolve_tac prems 1 

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ORELSE (resolve_tac [conjunct1, conjunct2] 1 THEN 

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resolve_tac prems 1))) ]); 

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val impE = prove_goal IFOL.thy 

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"[ P>Q; P; Q ==> R ] ==> R" 

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(fn prems=> [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]); 

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val allE = prove_goal IFOL.thy 

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"[ ALL x.P(x); P(x) ==> R ] ==> R" 

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(fn prems=> [ (REPEAT (resolve_tac (prems@[spec]) 1)) ]); 

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(*Duplicates the quantifier; for use with eresolve_tac*) 

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val all_dupE = prove_goal IFOL.thy 

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"[ ALL x.P(x); [ P(x); ALL x.P(x) ] ==> R \ 

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\ ] ==> R" 

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(fn prems=> [ (REPEAT (resolve_tac (prems@[spec]) 1)) ]); 

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(*** Negation rules, which translate between ~P and P>False ***) 

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val notI = prove_goalw IFOL.thy [not_def] "(P ==> False) ==> ~P" 

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(fn prems=> [ (REPEAT (ares_tac (prems@[impI]) 1)) ]); 

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val notE = prove_goalw IFOL.thy [not_def] "[ ~P; P ] ==> R" 

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(fn prems=> 

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[ (resolve_tac [mp RS FalseE] 1), 

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(REPEAT (resolve_tac prems 1)) ]); 

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(*This is useful with the special implication rules for each kind of P. *) 

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val not_to_imp = prove_goal IFOL.thy 

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"[ ~P; (P>False) ==> Q ] ==> Q" 

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(fn prems=> [ (REPEAT (ares_tac (prems@[impI,notE]) 1)) ]); 

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(* For substitution int an assumption P, reduce Q to P>Q, substitute into 

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this implication, then apply impI to move P back into the assumptions. 

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To specify P use something like 

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eres_inst_tac [ ("P","ALL y. ?S(x,y)") ] rev_mp 1 *) 

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val rev_mp = prove_goal IFOL.thy "[ P; P > Q ] ==> Q" 

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(fn prems=> [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]); 

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(*Contrapositive of an inference rule*) 

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val contrapos = prove_goal IFOL.thy "[ ~Q; P==>Q ] ==> ~P" 

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(fn [major,minor]=> 

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[ (rtac (major RS notE RS notI) 1), 

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(etac minor 1) ]); 

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(*** Modus Ponens Tactics ***) 

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(*Finds P>Q and P in the assumptions, replaces implication by Q *) 

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fun mp_tac i = eresolve_tac [notE,impE] i THEN assume_tac i; 

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(*Like mp_tac but instantiates no variables*) 

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fun eq_mp_tac i = eresolve_tac [notE,impE] i THEN eq_assume_tac i; 

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(*** Ifandonlyif ***) 

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val iffI = prove_goalw IFOL.thy [iff_def] 

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"[ P ==> Q; Q ==> P ] ==> P<>Q" 

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(fn prems=> [ (REPEAT (ares_tac (prems@[conjI, impI]) 1)) ]); 

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(*Observe use of rewrite_rule to unfold "<>" in metaassumptions (prems) *) 

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val iffE = prove_goalw IFOL.thy [iff_def] 

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"[ P <> Q; [ P>Q; Q>P ] ==> R ] ==> R" 

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(fn prems => [ (resolve_tac [conjE] 1), (REPEAT (ares_tac prems 1)) ]); 

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(* Destruct rules for <> similar to Modus Ponens *) 

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val iffD1 = prove_goalw IFOL.thy [iff_def] "[ P <> Q; P ] ==> Q" 

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(fn prems => [ (rtac (conjunct1 RS mp) 1), (REPEAT (ares_tac prems 1)) ]); 

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val iffD2 = prove_goalw IFOL.thy [iff_def] "[ P <> Q; Q ] ==> P" 

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(fn prems => [ (rtac (conjunct2 RS mp) 1), (REPEAT (ares_tac prems 1)) ]); 

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val iff_refl = prove_goal IFOL.thy "P <> P" 

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(fn _ => [ (REPEAT (ares_tac [iffI] 1)) ]); 

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val iff_sym = prove_goal IFOL.thy "Q <> P ==> P <> Q" 

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(fn [major] => 

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[ (rtac (major RS iffE) 1), 

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(rtac iffI 1), 

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(REPEAT (eresolve_tac [asm_rl,mp] 1)) ]); 

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val iff_trans = prove_goal IFOL.thy 

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"!!P Q R. [ P <> Q; Q<> R ] ==> P <> R" 

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(fn _ => 

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[ (rtac iffI 1), 

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(REPEAT (eresolve_tac [asm_rl,iffE] 1 ORELSE mp_tac 1)) ]); 

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(*** Unique existence. NOTE THAT the following 2 quantifications 

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EX!x such that [EX!y such that P(x,y)] (sequential) 

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EX!x,y such that P(x,y) (simultaneous) 

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do NOT mean the same thing. The parser treats EX!x y.P(x,y) as sequential. 

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***) 

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val ex1I = prove_goalw IFOL.thy [ex1_def] 

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"[ P(a); !!x. P(x) ==> x=a ] ==> EX! x. P(x)" 

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(fn prems => [ (REPEAT (ares_tac (prems@[exI,conjI,allI,impI]) 1)) ]); 

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(*Sometimes easier to use: the premises have no shared variables*) 
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val ex_ex1I = prove_goal IFOL.thy 
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"[ EX x.P(x); !!x y. [ P(x); P(y) ] ==> x=y ] ==> EX! x. P(x)" 
f17d542276b6
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changeset

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(fn [ex,eq] => [ (rtac (ex RS exE) 1), 
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(REPEAT (ares_tac [ex1I,eq] 1)) ]); 
f17d542276b6
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val ex1E = prove_goalw IFOL.thy [ex1_def] 
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"[ EX! x.P(x); !!x. [ P(x); ALL y. P(y) > y=x ] ==> R ] ==> R" 

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(fn prems => 

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[ (cut_facts_tac prems 1), 

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(REPEAT (eresolve_tac [exE,conjE] 1 ORELSE ares_tac prems 1)) ]); 

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(*** <> congruence rules for simplification ***) 

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(*Use iffE on a premise. For conj_cong, imp_cong, all_cong, ex_cong*) 

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fun iff_tac prems i = 

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resolve_tac (prems RL [iffE]) i THEN 

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REPEAT1 (eresolve_tac [asm_rl,mp] i); 

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val conj_cong = prove_goal IFOL.thy 

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"[ P <> P'; P' ==> Q <> Q' ] ==> (P&Q) <> (P'&Q')" 

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(fn prems => 

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[ (cut_facts_tac prems 1), 

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(REPEAT (ares_tac [iffI,conjI] 1 

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ORELSE eresolve_tac [iffE,conjE,mp] 1 

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ORELSE iff_tac prems 1)) ]); 

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val disj_cong = prove_goal IFOL.thy 

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"[ P <> P'; Q <> Q' ] ==> (PQ) <> (P'Q')" 

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(fn prems => 

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[ (cut_facts_tac prems 1), 

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(REPEAT (eresolve_tac [iffE,disjE,disjI1,disjI2] 1 

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ORELSE ares_tac [iffI] 1 

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ORELSE mp_tac 1)) ]); 

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val imp_cong = prove_goal IFOL.thy 

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"[ P <> P'; P' ==> Q <> Q' ] ==> (P>Q) <> (P'>Q')" 

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(fn prems => 

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[ (cut_facts_tac prems 1), 

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(REPEAT (ares_tac [iffI,impI] 1 

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ORELSE eresolve_tac [iffE] 1 

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ORELSE mp_tac 1 ORELSE iff_tac prems 1)) ]); 

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val iff_cong = prove_goal IFOL.thy 

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"[ P <> P'; Q <> Q' ] ==> (P<>Q) <> (P'<>Q')" 

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(fn prems => 

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[ (cut_facts_tac prems 1), 

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(REPEAT (eresolve_tac [iffE] 1 

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ORELSE ares_tac [iffI] 1 

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ORELSE mp_tac 1)) ]); 

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val not_cong = prove_goal IFOL.thy 

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"P <> P' ==> ~P <> ~P'" 

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(fn prems => 

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[ (cut_facts_tac prems 1), 

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(REPEAT (ares_tac [iffI,notI] 1 

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ORELSE mp_tac 1 

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ORELSE eresolve_tac [iffE,notE] 1)) ]); 

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val all_cong = prove_goal IFOL.thy 

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"(!!x.P(x) <> Q(x)) ==> (ALL x.P(x)) <> (ALL x.Q(x))" 

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(fn prems => 

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[ (REPEAT (ares_tac [iffI,allI] 1 

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ORELSE mp_tac 1 

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ORELSE eresolve_tac [allE] 1 ORELSE iff_tac prems 1)) ]); 

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val ex_cong = prove_goal IFOL.thy 

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"(!!x.P(x) <> Q(x)) ==> (EX x.P(x)) <> (EX x.Q(x))" 

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(fn prems => 

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[ (REPEAT (eresolve_tac [exE] 1 ORELSE ares_tac [iffI,exI] 1 

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ORELSE mp_tac 1 

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ORELSE iff_tac prems 1)) ]); 

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val ex1_cong = prove_goal IFOL.thy 

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"(!!x.P(x) <> Q(x)) ==> (EX! x.P(x)) <> (EX! x.Q(x))" 

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(fn prems => 

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[ (REPEAT (eresolve_tac [ex1E, spec RS mp] 1 ORELSE ares_tac [iffI,ex1I] 1 

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ORELSE mp_tac 1 

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ORELSE iff_tac prems 1)) ]); 

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(*** Equality rules ***) 

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val sym = prove_goal IFOL.thy "a=b ==> b=a" 

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(fn [major] => [ (rtac (major RS subst) 1), (rtac refl 1) ]); 

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val trans = prove_goal IFOL.thy "[ a=b; b=c ] ==> a=c" 

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(fn [prem1,prem2] => [ (rtac (prem2 RS subst) 1), (rtac prem1 1) ]); 

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(** ~ b=a ==> ~ a=b **) 

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val [not_sym] = compose(sym,2,contrapos); 

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(*calling "standard" reduces maxidx to 0*) 

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val ssubst = standard (sym RS subst); 

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(*A special case of ex1E that would otherwise need quantifier expansion*) 

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val ex1_equalsE = prove_goal IFOL.thy 

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"[ EX! x.P(x); P(a); P(b) ] ==> a=b" 

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(fn prems => 

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[ (cut_facts_tac prems 1), 

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(etac ex1E 1), 

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(rtac trans 1), 

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(rtac sym 2), 

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(REPEAT (eresolve_tac [asm_rl, spec RS mp] 1)) ]); 

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(** Polymorphic congruence rules **) 

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val subst_context = prove_goal IFOL.thy 

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"[ a=b ] ==> t(a)=t(b)" 

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(fn prems=> 

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[ (resolve_tac (prems RL [ssubst]) 1), 

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(resolve_tac [refl] 1) ]); 

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val subst_context2 = prove_goal IFOL.thy 

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"[ a=b; c=d ] ==> t(a,c)=t(b,d)" 

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(fn prems=> 

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[ (EVERY1 (map rtac ((prems RL [ssubst]) @ [refl]))) ]); 

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val subst_context3 = prove_goal IFOL.thy 

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"[ a=b; c=d; e=f ] ==> t(a,c,e)=t(b,d,f)" 

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(fn prems=> 

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[ (EVERY1 (map rtac ((prems RL [ssubst]) @ [refl]))) ]); 

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(*Useful with eresolve_tac for proving equalties from known equalities. 

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a = b 

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c = d *) 

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val box_equals = prove_goal IFOL.thy 

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"[ a=b; a=c; b=d ] ==> c=d" 

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(fn prems=> 

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[ (resolve_tac [trans] 1), 

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(resolve_tac [trans] 1), 

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(resolve_tac [sym] 1), 

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(REPEAT (resolve_tac prems 1)) ]); 

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(*Dual of box_equals: for proving equalities backwards*) 

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val simp_equals = prove_goal IFOL.thy 

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"[ a=c; b=d; c=d ] ==> a=b" 

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(fn prems=> 

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[ (resolve_tac [trans] 1), 

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(resolve_tac [trans] 1), 

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(REPEAT (resolve_tac (prems @ (prems RL [sym])) 1)) ]); 

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(** Congruence rules for predicate letters **) 

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val pred1_cong = prove_goal IFOL.thy 

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"a=a' ==> P(a) <> P(a')" 

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(fn prems => 

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[ (cut_facts_tac prems 1), 

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(rtac iffI 1), 

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(DEPTH_SOLVE (eresolve_tac [asm_rl, subst, ssubst] 1)) ]); 

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val pred2_cong = prove_goal IFOL.thy 

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"[ a=a'; b=b' ] ==> P(a,b) <> P(a',b')" 

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(fn prems => 

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[ (cut_facts_tac prems 1), 

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(rtac iffI 1), 

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(DEPTH_SOLVE (eresolve_tac [asm_rl, subst, ssubst] 1)) ]); 

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val pred3_cong = prove_goal IFOL.thy 

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"[ a=a'; b=b'; c=c' ] ==> P(a,b,c) <> P(a',b',c')" 

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(fn prems => 

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[ (cut_facts_tac prems 1), 

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(rtac iffI 1), 

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(DEPTH_SOLVE (eresolve_tac [asm_rl, subst, ssubst] 1)) ]); 

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(*special cases for free variables P, Q, R, S  up to 3 arguments*) 

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val pred_congs = 

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flat (map (fn c => 

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map (fn th => read_instantiate [("P",c)] th) 

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[pred1_cong,pred2_cong,pred3_cong]) 

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(explode"PQRS")); 

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(*special case for the equality predicate!*) 

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val eq_cong = read_instantiate [("P","op =")] pred2_cong; 

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(*** Simplifications of assumed implications. 

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Roy Dyckhoff has proved that conj_impE, disj_impE, and imp_impE 

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used with mp_tac (restricted to atomic formulae) is COMPLETE for 

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intuitionistic propositional logic. See 

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R. Dyckhoff, Contractionfree sequent calculi for intuitionistic logic 

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(preprint, University of St Andrews, 1991) ***) 

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val conj_impE = prove_goal IFOL.thy 

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"[ (P&Q)>S; P>(Q>S) ==> R ] ==> R" 

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(fn major::prems=> 

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[ (REPEAT (ares_tac ([conjI, impI, major RS mp]@prems) 1)) ]); 

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val disj_impE = prove_goal IFOL.thy 

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"[ (PQ)>S; [ P>S; Q>S ] ==> R ] ==> R" 

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(fn major::prems=> 

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[ (DEPTH_SOLVE (ares_tac ([disjI1, disjI2, impI, major RS mp]@prems) 1)) ]); 

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(*Simplifies the implication. Classical version is stronger. 

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Still UNSAFE since Q must be provable  backtracking needed. *) 

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val imp_impE = prove_goal IFOL.thy 

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"[ (P>Q)>S; [ P; Q>S ] ==> Q; S ==> R ] ==> R" 

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(fn major::prems=> 

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[ (REPEAT (ares_tac ([impI, major RS mp]@prems) 1)) ]); 

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(*Simplifies the implication. Classical version is stronger. 

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Still UNSAFE since ~P must be provable  backtracking needed. *) 

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val not_impE = prove_goal IFOL.thy 

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"[ ~P > S; P ==> False; S ==> R ] ==> R" 

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(fn major::prems=> 

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[ (REPEAT (ares_tac ([notI, impI, major RS mp]@prems) 1)) ]); 

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(*Simplifies the implication. UNSAFE. *) 

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val iff_impE = prove_goal IFOL.thy 

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"[ (P<>Q)>S; [ P; Q>S ] ==> Q; [ Q; P>S ] ==> P; \ 

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\ S ==> R ] ==> R" 

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(fn major::prems=> 

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[ (REPEAT (ares_tac ([iffI, impI, major RS mp]@prems) 1)) ]); 

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(*What if (ALL x.~~P(x)) > ~~(ALL x.P(x)) is an assumption? UNSAFE*) 

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val all_impE = prove_goal IFOL.thy 

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"[ (ALL x.P(x))>S; !!x.P(x); S ==> R ] ==> R" 

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(fn major::prems=> 

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[ (REPEAT (ares_tac ([allI, impI, major RS mp]@prems) 1)) ]); 

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(*Unsafe: (EX x.P(x))>S is equivalent to ALL x.P(x)>S. *) 

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val ex_impE = prove_goal IFOL.thy 

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"[ (EX x.P(x))>S; P(x)>S ==> R ] ==> R" 

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(fn major::prems=> 

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[ (REPEAT (ares_tac ([exI, impI, major RS mp]@prems) 1)) ]); 

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end; 

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open IFOL_Lemmas; 

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