author  hoelzl 
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parent 61386  0a29a984a91b 
child 69593  3dda49e08b9d 
permissions  rwrr 
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(* Title: Sequents/LK0.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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There may be printing problems if a seqent is in expanded normal form 

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(etaexpanded, betacontracted). 
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*) 
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section \<open>Classical FirstOrder Sequent Calculus\<close> 
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theory LK0 

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imports Sequents 

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begin 

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class "term" 
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default_sort "term" 
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consts 

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Trueprop :: "two_seqi" 
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True :: o 
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False :: o 

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equal :: "['a,'a] \<Rightarrow> o" (infixl "=" 50) 
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Not :: "o \<Rightarrow> o" ("\<not> _" [40] 40) 

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conj :: "[o,o] \<Rightarrow> o" (infixr "\<and>" 35) 

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disj :: "[o,o] \<Rightarrow> o" (infixr "\<or>" 30) 

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imp :: "[o,o] \<Rightarrow> o" (infixr "\<longrightarrow>" 25) 

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iff :: "[o,o] \<Rightarrow> o" (infixr "\<longleftrightarrow>" 25) 

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The :: "('a \<Rightarrow> o) \<Rightarrow> 'a" (binder "THE " 10) 

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All :: "('a \<Rightarrow> o) \<Rightarrow> o" (binder "\<forall>" 10) 

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Ex :: "('a \<Rightarrow> o) \<Rightarrow> o" (binder "\<exists>" 10) 

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syntax 

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"_Trueprop" :: "two_seqe" ("((_)/ \<turnstile> (_))" [6,6] 5) 
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parse_translation \<open>[(@{syntax_const "_Trueprop"}, K (two_seq_tr @{const_syntax Trueprop}))]\<close> 
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print_translation \<open>[(@{const_syntax Trueprop}, K (two_seq_tr' @{syntax_const "_Trueprop"}))]\<close> 

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abbreviation 
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not_equal (infixl "\<noteq>" 50) where 
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"x \<noteq> y \<equiv> \<not> (x = y)" 

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axiomatization where 
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(*Structural rules: contraction, thinning, exchange [Soren Heilmann] *) 

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contRS: "$H \<turnstile> $E, $S, $S, $F \<Longrightarrow> $H \<turnstile> $E, $S, $F" and 
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contLS: "$H, $S, $S, $G \<turnstile> $E \<Longrightarrow> $H, $S, $G \<turnstile> $E" and 

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thinRS: "$H \<turnstile> $E, $F \<Longrightarrow> $H \<turnstile> $E, $S, $F" and 
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thinLS: "$H, $G \<turnstile> $E \<Longrightarrow> $H, $S, $G \<turnstile> $E" and 

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exchRS: "$H \<turnstile> $E, $R, $S, $F \<Longrightarrow> $H \<turnstile> $E, $S, $R, $F" and 
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exchLS: "$H, $R, $S, $G \<turnstile> $E \<Longrightarrow> $H, $S, $R, $G \<turnstile> $E" and 

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cut: "\<lbrakk>$H \<turnstile> $E, P; $H, P \<turnstile> $E\<rbrakk> \<Longrightarrow> $H \<turnstile> $E" and 
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(*Propositional rules*) 

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basic: "$H, P, $G \<turnstile> $E, P, $F" and 
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conjR: "\<lbrakk>$H\<turnstile> $E, P, $F; $H\<turnstile> $E, Q, $F\<rbrakk> \<Longrightarrow> $H\<turnstile> $E, P \<and> Q, $F" and 
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conjL: "$H, P, Q, $G \<turnstile> $E \<Longrightarrow> $H, P \<and> Q, $G \<turnstile> $E" and 

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disjR: "$H \<turnstile> $E, P, Q, $F \<Longrightarrow> $H \<turnstile> $E, P \<or> Q, $F" and 
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disjL: "\<lbrakk>$H, P, $G \<turnstile> $E; $H, Q, $G \<turnstile> $E\<rbrakk> \<Longrightarrow> $H, P \<or> Q, $G \<turnstile> $E" and 

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impR: "$H, P \<turnstile> $E, Q, $F \<Longrightarrow> $H \<turnstile> $E, P \<longrightarrow> Q, $F" and 
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impL: "\<lbrakk>$H,$G \<turnstile> $E,P; $H, Q, $G \<turnstile> $E\<rbrakk> \<Longrightarrow> $H, P \<longrightarrow> Q, $G \<turnstile> $E" and 

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notR: "$H, P \<turnstile> $E, $F \<Longrightarrow> $H \<turnstile> $E, \<not> P, $F" and 
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notL: "$H, $G \<turnstile> $E, P \<Longrightarrow> $H, \<not> P, $G \<turnstile> $E" and 

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FalseL: "$H, False, $G \<turnstile> $E" and 
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True_def: "True \<equiv> False \<longrightarrow> False" and 
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iff_def: "P \<longleftrightarrow> Q \<equiv> (P \<longrightarrow> Q) \<and> (Q \<longrightarrow> P)" 

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axiomatization where 
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(*Quantifiers*) 
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allR: "(\<And>x. $H \<turnstile> $E, P(x), $F) \<Longrightarrow> $H \<turnstile> $E, \<forall>x. P(x), $F" and 
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allL: "$H, P(x), $G, \<forall>x. P(x) \<turnstile> $E \<Longrightarrow> $H, \<forall>x. P(x), $G \<turnstile> $E" and 

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exR: "$H \<turnstile> $E, P(x), $F, \<exists>x. P(x) \<Longrightarrow> $H \<turnstile> $E, \<exists>x. P(x), $F" and 
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exL: "(\<And>x. $H, P(x), $G \<turnstile> $E) \<Longrightarrow> $H, \<exists>x. P(x), $G \<turnstile> $E" and 

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(*Equality*) 

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refl: "$H \<turnstile> $E, a = a, $F" and 
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subst: "\<And>G H E. $H(a), $G(a) \<turnstile> $E(a) \<Longrightarrow> $H(b), a=b, $G(b) \<turnstile> $E(b)" 

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(* Reflection *) 

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axiomatization where 
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eq_reflection: "\<turnstile> x = y \<Longrightarrow> (x \<equiv> y)" and 
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iff_reflection: "\<turnstile> P \<longleftrightarrow> Q \<Longrightarrow> (P \<equiv> Q)" 

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(*Descriptions*) 

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axiomatization where 
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The: "\<lbrakk>$H \<turnstile> $E, P(a), $F; \<And>x.$H, P(x) \<turnstile> $E, x=a, $F\<rbrakk> \<Longrightarrow> 
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$H \<turnstile> $E, P(THE x. P(x)), $F" 

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definition If :: "[o, 'a, 'a] \<Rightarrow> 'a" ("(if (_)/ then (_)/ else (_))" 10) 
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where "If(P,x,y) \<equiv> THE z::'a. (P \<longrightarrow> z = x) \<and> (\<not> P \<longrightarrow> z = y)" 

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(** Structural Rules on formulas **) 

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(*contraction*) 

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lemma contR: "$H \<turnstile> $E, P, P, $F \<Longrightarrow> $H \<turnstile> $E, P, $F" 
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by (rule contRS) 
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lemma contL: "$H, P, P, $G \<turnstile> $E \<Longrightarrow> $H, P, $G \<turnstile> $E" 
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by (rule contLS) 
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(*thinning*) 

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lemma thinR: "$H \<turnstile> $E, $F \<Longrightarrow> $H \<turnstile> $E, P, $F" 
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by (rule thinRS) 
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lemma thinL: "$H, $G \<turnstile> $E \<Longrightarrow> $H, P, $G \<turnstile> $E" 
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by (rule thinLS) 
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(*exchange*) 

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lemma exchR: "$H \<turnstile> $E, Q, P, $F \<Longrightarrow> $H \<turnstile> $E, P, Q, $F" 
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by (rule exchRS) 
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lemma exchL: "$H, Q, P, $G \<turnstile> $E \<Longrightarrow> $H, P, Q, $G \<turnstile> $E" 
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by (rule exchLS) 
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ML \<open> 
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(*Cut and thin, replacing the rightside formula*) 
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fun cutR_tac ctxt s i = 
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Rule_Insts.res_inst_tac ctxt [((("P", 0), Position.none), s)] [] @{thm cut} i THEN 
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resolve_tac ctxt @{thms thinR} i 
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(*Cut and thin, replacing the leftside formula*) 

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fun cutL_tac ctxt s i = 
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Rule_Insts.res_inst_tac ctxt [((("P", 0), Position.none), s)] [] @{thm cut} i THEN 
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resolve_tac ctxt @{thms thinL} (i + 1) 
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\<close> 
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(** Ifandonlyif rules **) 

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lemma iffR: "\<lbrakk>$H,P \<turnstile> $E,Q,$F; $H,Q \<turnstile> $E,P,$F\<rbrakk> \<Longrightarrow> $H \<turnstile> $E, P \<longleftrightarrow> Q, $F" 
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apply (unfold iff_def) 
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apply (assumption  rule conjR impR)+ 

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done 

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lemma iffL: "\<lbrakk>$H,$G \<turnstile> $E,P,Q; $H,Q,P,$G \<turnstile> $E\<rbrakk> \<Longrightarrow> $H, P \<longleftrightarrow> Q, $G \<turnstile> $E" 
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apply (unfold iff_def) 
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apply (assumption  rule conjL impL basic)+ 

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done 

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lemma iff_refl: "$H \<turnstile> $E, (P \<longleftrightarrow> P), $F" 
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apply (rule iffR basic)+ 
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done 

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lemma TrueR: "$H \<turnstile> $E, True, $F" 
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apply (unfold True_def) 
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apply (rule impR) 

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apply (rule basic) 

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done 

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(*Descriptions*) 

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lemma the_equality: 

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assumes p1: "$H \<turnstile> $E, P(a), $F" 
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and p2: "\<And>x. $H, P(x) \<turnstile> $E, x=a, $F" 

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shows "$H \<turnstile> $E, (THE x. P(x)) = a, $F" 

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apply (rule cut) 
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apply (rule_tac [2] p2) 

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apply (rule The, rule thinR, rule exchRS, rule p1) 

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apply (rule thinR, rule exchRS, rule p2) 

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done 

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(** Weakened quantifier rules. Incomplete, they let the search terminate.**) 

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lemma allL_thin: "$H, P(x), $G \<turnstile> $E \<Longrightarrow> $H, \<forall>x. P(x), $G \<turnstile> $E" 
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apply (rule allL) 
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apply (erule thinL) 

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done 

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lemma exR_thin: "$H \<turnstile> $E, P(x), $F \<Longrightarrow> $H \<turnstile> $E, \<exists>x. P(x), $F" 
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apply (rule exR) 
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apply (erule thinR) 

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done 

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(*The rules of LK*) 

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lemmas [safe] = 
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iffR iffL 

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notR notL 

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impR impL 

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disjR disjL 

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conjR conjL 

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FalseL TrueR 

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refl basic 

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ML \<open>val prop_pack = Cla.get_pack @{context}\<close> 
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lemmas [safe] = exL allR 

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lemmas [unsafe] = the_equality exR_thin allL_thin 

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ML \<open>val LK_pack = Cla.get_pack @{context}\<close> 
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ML \<open> 
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val LK_dup_pack = 
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Cla.put_pack prop_pack @{context} 

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> fold_rev Cla.add_safe @{thms allR exL} 

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> fold_rev Cla.add_unsafe @{thms allL exR the_equality} 

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> Cla.get_pack; 

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\<close> 
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method_setup fast_prop = 
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\<open>Scan.succeed (fn ctxt => SIMPLE_METHOD' (Cla.fast_tac (Cla.put_pack prop_pack ctxt)))\<close> 
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method_setup fast_dup = 
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\<open>Scan.succeed (fn ctxt => SIMPLE_METHOD' (Cla.fast_tac (Cla.put_pack LK_dup_pack ctxt)))\<close> 
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method_setup best_dup = 

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\<open>Scan.succeed (fn ctxt => SIMPLE_METHOD' (Cla.best_tac (Cla.put_pack LK_dup_pack ctxt)))\<close> 
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method_setup lem = \<open> 
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Attrib.thm >> (fn th => fn ctxt => 
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SIMPLE_METHOD' (fn i => 
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resolve_tac ctxt [@{thm thinR} RS @{thm cut}] i THEN 
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REPEAT (resolve_tac ctxt @{thms thinL} i) THEN 

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resolve_tac ctxt [th] i)) 

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\<close> 
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lemma mp_R: 
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assumes major: "$H \<turnstile> $E, $F, P \<longrightarrow> Q" 
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and minor: "$H \<turnstile> $E, $F, P" 

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shows "$H \<turnstile> $E, Q, $F" 

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apply (rule thinRS [THEN cut], rule major) 
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apply step 
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apply (rule thinR, rule minor) 
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done 

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lemma mp_L: 

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assumes major: "$H, $G \<turnstile> $E, P \<longrightarrow> Q" 
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and minor: "$H, $G, Q \<turnstile> $E" 

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shows "$H, P, $G \<turnstile> $E" 

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apply (rule thinL [THEN cut], rule major) 
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apply step 
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apply (rule thinL, rule minor) 
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done 

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(** Two rules to generate left and right rules from implications **) 

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lemma R_of_imp: 

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assumes major: "\<turnstile> P \<longrightarrow> Q" 
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and minor: "$H \<turnstile> $E, $F, P" 

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shows "$H \<turnstile> $E, Q, $F" 

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apply (rule mp_R) 
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apply (rule_tac [2] minor) 

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apply (rule thinRS, rule major [THEN thinLS]) 

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done 

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lemma L_of_imp: 

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assumes major: "\<turnstile> P \<longrightarrow> Q" 
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and minor: "$H, $G, Q \<turnstile> $E" 

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shows "$H, P, $G \<turnstile> $E" 

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apply (rule mp_L) 
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apply (rule_tac [2] minor) 

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apply (rule thinRS, rule major [THEN thinLS]) 

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done 

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(*Can be used to create implications in a subgoal*) 

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lemma backwards_impR: 

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assumes prem: "$H, $G \<turnstile> $E, $F, P \<longrightarrow> Q" 
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shows "$H, P, $G \<turnstile> $E, Q, $F" 

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apply (rule mp_L) 
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apply (rule_tac [2] basic) 

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apply (rule thinR, rule prem) 

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done 

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lemma conjunct1: "\<turnstile>P \<and> Q \<Longrightarrow> \<turnstile>P" 
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apply (erule thinR [THEN cut]) 
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apply fast 

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done 

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lemma conjunct2: "\<turnstile>P \<and> Q \<Longrightarrow> \<turnstile>Q" 
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apply (erule thinR [THEN cut]) 
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apply fast 

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done 

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lemma spec: "\<turnstile> (\<forall>x. P(x)) \<Longrightarrow> \<turnstile> P(x)" 
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apply (erule thinR [THEN cut]) 
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apply fast 

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done 

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(** Equality **) 

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lemma sym: "\<turnstile> a = b \<longrightarrow> b = a" 
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by (safe add!: subst) 
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lemma trans: "\<turnstile> a = b \<longrightarrow> b = c \<longrightarrow> a = c" 
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by (safe add!: subst) 
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(* Symmetry of equality in hypotheses *) 

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lemmas symL = sym [THEN L_of_imp] 
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(* Symmetry of equality in hypotheses *) 

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lemmas symR = sym [THEN R_of_imp] 
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lemma transR: "\<lbrakk>$H\<turnstile> $E, $F, a = b; $H\<turnstile> $E, $F, b=c\<rbrakk> \<Longrightarrow> $H\<turnstile> $E, a = c, $F" 
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by (rule trans [THEN R_of_imp, THEN mp_R]) 
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(* Two theorms for rewriting only one instance of a definition: 

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the first for definitions of formulae and the second for terms *) 

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lemma def_imp_iff: "(A \<equiv> B) \<Longrightarrow> \<turnstile> A \<longleftrightarrow> B" 
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apply unfold 
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apply (rule iff_refl) 

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done 

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lemma meta_eq_to_obj_eq: "(A \<equiv> B) \<Longrightarrow> \<turnstile> A = B" 
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apply unfold 
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apply (rule refl) 

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done 

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(** ifthenelse rules **) 

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lemma if_True: "\<turnstile> (if True then x else y) = x" 
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unfolding If_def by fast 
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lemma if_False: "\<turnstile> (if False then x else y) = y" 
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unfolding If_def by fast 
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lemma if_P: "\<turnstile> P \<Longrightarrow> \<turnstile> (if P then x else y) = x" 
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apply (unfold If_def) 
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apply (erule thinR [THEN cut]) 

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apply fast 

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done 

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lemma if_not_P: "\<turnstile> \<not> P \<Longrightarrow> \<turnstile> (if P then x else y) = y" 
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apply (unfold If_def) 
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apply (erule thinR [THEN cut]) 

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apply fast 

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done 

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end 