author  wenzelm 
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(* Title: Sequents/LK.thy 
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory 
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Copyright 1993 University of Cambridge 
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Axiom to express monotonicity (a variant of the deduction theorem). Makes the 
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link between  and ==>, needed for instance to prove imp_cong. 
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Axiom left_cong allows the simplifier to use leftside formulas. Ideally it 
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should be derived from lowerlevel axioms. 

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CANNOT be added to LK0.thy because modal logic is built upon it, and 
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various modal rules would become inconsistent. 
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*) 
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17481  15 
theory LK 
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imports LK0 

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uses ("simpdata.ML") 

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begin 

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axiomatization where 
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monotonic: "($H  P ==> $H  Q) ==> $H, P  Q" and 

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17481  23 
left_cong: "[ P == P';  P' ==> ($H  $F) == ($H'  $F') ] 
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==> (P, $H  $F) == (P', $H'  $F')" 

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27149  26 

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subsection {* Rewrite rules *} 

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lemma conj_simps: 

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" P & True <> P" 

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" True & P <> P" 

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" P & False <> False" 

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" False & P <> False" 

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" P & P <> P" 

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" P & P & Q <> P & Q" 

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" P & ~P <> False" 

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" ~P & P <> False" 

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" (P & Q) & R <> P & (Q & R)" 

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apply (tactic {* ALLGOALS (fast_tac (LK_pack add_safes @{thms subst})) *}) 

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done 

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lemma disj_simps: 

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" P  True <> True" 

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" True  P <> True" 

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" P  False <> P" 

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" False  P <> P" 

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" P  P <> P" 

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" P  P  Q <> P  Q" 

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" (P  Q)  R <> P  (Q  R)" 

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apply (tactic {* ALLGOALS (fast_tac (LK_pack add_safes @{thms subst})) *}) 

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done 

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lemma not_simps: 

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" ~ False <> True" 

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" ~ True <> False" 

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apply (tactic {* ALLGOALS (fast_tac (LK_pack add_safes @{thms subst})) *}) 

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done 

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lemma imp_simps: 

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" (P > False) <> ~P" 

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" (P > True) <> True" 

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" (False > P) <> True" 

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" (True > P) <> P" 

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" (P > P) <> True" 

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" (P > ~P) <> ~P" 

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apply (tactic {* ALLGOALS (fast_tac (LK_pack add_safes @{thms subst})) *}) 

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done 

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lemma iff_simps: 

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" (True <> P) <> P" 

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" (P <> True) <> P" 

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" (P <> P) <> True" 

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" (False <> P) <> ~P" 

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" (P <> False) <> ~P" 

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apply (tactic {* ALLGOALS (fast_tac (LK_pack add_safes @{thms subst})) *}) 

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done 

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lemma quant_simps: 

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

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

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

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

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"!!P.  (EX x. x=t & P(x)) <> P(t)" 

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"!!P.  (EX x. t=x & P(x)) <> P(t)" 

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apply (tactic {* ALLGOALS (fast_tac (LK_pack add_safes @{thms subst})) *}) 

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done 

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subsection {* Miniscoping: pushing quantifiers in *} 

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text {* 

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We do NOT distribute of ALL over &, or dually that of EX over  

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Baaz and Leitsch, On Skolemization and Proof Complexity (1994) 

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show that this step can increase proof length! 

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*} 

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text {*existential miniscoping*} 

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lemma ex_simps: 

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

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

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

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

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

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

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apply (tactic {* ALLGOALS (fast_tac (LK_pack add_safes @{thms subst})) *}) 

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done 

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text {*universal miniscoping*} 

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lemma all_simps: 

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

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

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

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

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

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

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apply (tactic {* ALLGOALS (fast_tac (LK_pack add_safes @{thms subst})) *}) 

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done 

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text {*These are NOT supplied by default!*} 

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lemma distrib_simps: 

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" P & (Q  R) <> P&Q  P&R" 

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" (Q  R) & P <> Q&P  R&P" 

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" (P  Q > R) <> (P > R) & (Q > R)" 

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apply (tactic {* ALLGOALS (fast_tac (LK_pack add_safes @{thms subst})) *}) 

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done 

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lemma P_iff_F: " ~P ==>  (P <> False)" 

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apply (erule thinR [THEN cut]) 

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apply (tactic {* fast_tac LK_pack 1 *}) 

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done 

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lemmas iff_reflection_F = P_iff_F [THEN iff_reflection] 
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lemma P_iff_T: " P ==>  (P <> True)" 

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apply (erule thinR [THEN cut]) 

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apply (tactic {* fast_tac LK_pack 1 *}) 

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done 

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lemmas iff_reflection_T = P_iff_T [THEN iff_reflection] 
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lemma LK_extra_simps: 

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" P  ~P" 

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" ~P  P" 

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" ~ ~ P <> P" 

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" (~P > P) <> P" 

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" (~P <> ~Q) <> (P<>Q)" 

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apply (tactic {* ALLGOALS (fast_tac (LK_pack add_safes @{thms subst})) *}) 

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done 

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subsection {* Named rewrite rules *} 

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lemma conj_commute: " P&Q <> Q&P" 

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and conj_left_commute: " P&(Q&R) <> Q&(P&R)" 

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apply (tactic {* ALLGOALS (fast_tac (LK_pack add_safes @{thms subst})) *}) 

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done 

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lemmas conj_comms = conj_commute conj_left_commute 

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lemma disj_commute: " PQ <> QP" 

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and disj_left_commute: " P(QR) <> Q(PR)" 

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apply (tactic {* ALLGOALS (fast_tac (LK_pack add_safes @{thms subst})) *}) 

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done 

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lemmas disj_comms = disj_commute disj_left_commute 

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lemma conj_disj_distribL: " P&(QR) <> (P&Q  P&R)" 

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and conj_disj_distribR: " (PQ)&R <> (P&R  Q&R)" 

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and disj_conj_distribL: " P(Q&R) <> (PQ) & (PR)" 

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and disj_conj_distribR: " (P&Q)R <> (PR) & (QR)" 

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and imp_conj_distrib: " (P > (Q&R)) <> (P>Q) & (P>R)" 

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and imp_conj: " ((P&Q)>R) <> (P > (Q > R))" 

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and imp_disj: " (PQ > R) <> (P>R) & (Q>R)" 

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and imp_disj1: " (P>Q)  R <> (P>Q  R)" 

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and imp_disj2: " Q  (P>R) <> (P>Q  R)" 

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and de_Morgan_disj: " (~(P  Q)) <> (~P & ~Q)" 

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and de_Morgan_conj: " (~(P & Q)) <> (~P  ~Q)" 

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and not_iff: " ~(P <> Q) <> (P <> ~Q)" 

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apply (tactic {* ALLGOALS (fast_tac (LK_pack add_safes @{thms subst})) *}) 

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done 

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lemma imp_cong: 

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assumes p1: " P <> P'" 

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and p2: " P' ==>  Q <> Q'" 

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shows " (P>Q) <> (P'>Q')" 

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apply (tactic {* lemma_tac @{thm p1} 1 *}) 

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apply (tactic {* safe_tac LK_pack 1 *}) 

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apply (tactic {* 

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REPEAT (rtac @{thm cut} 1 THEN 

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DEPTH_SOLVE_1 

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(resolve_tac [@{thm thinL}, @{thm thinR}, @{thm p2} COMP @{thm monotonic}] 1) THEN 

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safe_tac LK_pack 1) *}) 

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done 

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lemma conj_cong: 

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assumes p1: " P <> P'" 

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and p2: " P' ==>  Q <> Q'" 

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shows " (P&Q) <> (P'&Q')" 

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apply (tactic {* lemma_tac @{thm p1} 1 *}) 

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apply (tactic {* safe_tac LK_pack 1 *}) 

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apply (tactic {* 

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REPEAT (rtac @{thm cut} 1 THEN 

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DEPTH_SOLVE_1 

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(resolve_tac [@{thm thinL}, @{thm thinR}, @{thm p2} COMP @{thm monotonic}] 1) THEN 

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safe_tac LK_pack 1) *}) 

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done 

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lemma eq_sym_conv: " (x=y) <> (y=x)" 

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apply (tactic {* fast_tac (LK_pack add_safes @{thms subst}) 1 *}) 

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done 

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use "simpdata.ML" 
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setup {* Simplifier.map_simpset_global (K LK_ss) *} 
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text {* To create substition rules *} 

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lemma eq_imp_subst: " a=b ==> $H, A(a), $G  $E, A(b), $F" 

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apply (tactic {* asm_simp_tac LK_basic_ss 1 *}) 

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done 

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lemma split_if: " P(if Q then x else y) <> ((Q > P(x)) & (~Q > P(y)))" 

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apply (rule_tac P = Q in cut) 

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apply (tactic {* simp_tac (@{simpset} addsimps @{thms if_P}) 2 *}) 
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apply (rule_tac P = "~Q" in cut) 
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apply (tactic {* simp_tac (@{simpset} addsimps @{thms if_not_P}) 2 *}) 
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apply (tactic {* fast_tac LK_pack 1 *}) 
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done 

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lemma if_cancel: " (if P then x else x) = x" 

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apply (tactic {* lemma_tac @{thm split_if} 1 *}) 

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apply (tactic {* fast_tac LK_pack 1 *}) 

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done 

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lemma if_eq_cancel: " (if x=y then y else x) = x" 

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apply (tactic {* lemma_tac @{thm split_if} 1 *}) 

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apply (tactic {* safe_tac LK_pack 1 *}) 

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apply (rule symL) 

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apply (rule basic) 

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done 

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end 