author  kuncar 
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permissions  rwrr 
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(* Title: HOL/Tools/transfer.ML 
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Author: Brian Huffman, TU Muenchen 

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Generic theorem transfer method. 

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*) 

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signature TRANSFER = 

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sig 

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val prep_conv: conv 

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val get_relator_eq: Proof.context > thm list 
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val get_sym_relator_eq: Proof.context > thm list 
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val get_transfer_raw: Proof.context > thm list 
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val transfer_add: attribute 
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val transfer_del: attribute 

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val transfer_rule_of_term: Proof.context > term > thm 
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val transfer_tac: bool > Proof.context > int > tactic 
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val transfer_prover_tac: Proof.context > int > tactic 
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val setup: theory > theory 
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end 

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structure Transfer : TRANSFER = 

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struct 

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(** Theory Data **) 
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structure Data = Generic_Data 
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( 
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type T = 
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{ transfer_raw : thm Item_Net.T, 
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known_frees : (string * typ) list, 
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compound_rhs : unit Net.net, 
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relator_eq : thm Item_Net.T, 
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relator_eq_raw : thm Item_Net.T } 
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val empty = 
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{ transfer_raw = Thm.full_rules, 
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known_frees = [], 
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compound_rhs = Net.empty, 
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relator_eq = Thm.full_rules, 
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relator_eq_raw = Thm.full_rules } 
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val extend = I 
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fun merge 
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( { transfer_raw = t1, known_frees = k1, 
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compound_rhs = c1, relator_eq = r1, 
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relator_eq_raw = rw1 }, 
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{ transfer_raw = t2, known_frees = k2, 
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compound_rhs = c2, relator_eq = r2, 
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relator_eq_raw = rw2 } ) = 
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{ transfer_raw = Item_Net.merge (t1, t2), 
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known_frees = Library.merge (op =) (k1, k2), 
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compound_rhs = Net.merge (K true) (c1, c2), 
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relator_eq = Item_Net.merge (r1, r2), 
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relator_eq_raw = Item_Net.merge (rw1, rw2) } 
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) 
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fun get_relator_eq ctxt = ctxt 
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> (Item_Net.content o #relator_eq o Data.get o Context.Proof) 
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> map safe_mk_meta_eq 
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fun get_sym_relator_eq ctxt = ctxt 
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> (Item_Net.content o #relator_eq o Data.get o Context.Proof) 
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> map (Thm.symmetric o safe_mk_meta_eq) 
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fun get_relator_eq_raw ctxt = ctxt 
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> (Item_Net.content o #relator_eq_raw o Data.get o Context.Proof) 
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fun get_transfer_raw ctxt = ctxt 
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> (Item_Net.content o #transfer_raw o Data.get o Context.Proof) 
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fun get_known_frees ctxt = ctxt 
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> (#known_frees o Data.get o Context.Proof) 
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fun get_compound_rhs ctxt = ctxt 
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> (#compound_rhs o Data.get o Context.Proof) 
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fun map_data f1 f2 f3 f4 f5 
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{ transfer_raw, known_frees, compound_rhs, relator_eq, relator_eq_raw } = 
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{ transfer_raw = f1 transfer_raw, 
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known_frees = f2 known_frees, 
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compound_rhs = f3 compound_rhs, 
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relator_eq = f4 relator_eq, 
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relator_eq_raw = f5 relator_eq_raw } 
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fun map_transfer_raw f = map_data f I I I I 
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fun map_known_frees f = map_data I f I I I 
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fun map_compound_rhs f = map_data I I f I I 
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fun map_relator_eq f = map_data I I I f I 
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fun map_relator_eq_raw f = map_data I I I I f 
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fun add_transfer_thm thm = Data.map 
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(map_transfer_raw (Item_Net.update thm) o 
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map_compound_rhs 
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(case HOLogic.dest_Trueprop (Thm.concl_of thm) of 
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_ $ _ $ (rhs as (_ $ _)) => Net.insert_term (K true) (rhs, ()) 
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 _ => I) o 
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map_known_frees (Term.add_frees (Thm.concl_of thm))) 
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fun del_transfer_thm thm = Data.map (map_transfer_raw (Item_Net.remove thm)) 
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(** Conversions **) 
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val Rel_rule = Thm.symmetric @{thm Rel_def} 

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fun dest_funcT cT = 

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(case Thm.dest_ctyp cT of [T, U] => (T, U) 

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 _ => raise TYPE ("dest_funcT", [Thm.typ_of cT], [])) 

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fun Rel_conv ct = 

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let val (cT, cT') = dest_funcT (Thm.ctyp_of_term ct) 

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val (cU, _) = dest_funcT cT' 

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in Drule.instantiate' [SOME cT, SOME cU] [SOME ct] Rel_rule end 

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(* Conversion to preprocess a transfer rule *) 
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fun prep_conv ct = ( 
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Conv.implies_conv Conv.all_conv prep_conv 

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else_conv 

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HOLogic.Trueprop_conv (Conv.fun_conv (Conv.fun_conv Rel_conv)) 
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else_conv 
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Conv.all_conv) ct 

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(** Replacing explicit equalities with is_equality premises **) 
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fun mk_is_equality t = 
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Const (@{const_name is_equality}, Term.fastype_of t > HOLogic.boolT) $ t 
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val is_equality_lemma = 
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@{lemma "(!!R. is_equality R ==> PROP (P R)) == PROP (P (op =))" 
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by (unfold is_equality_def, rule, drule meta_spec, 
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erule meta_mp, rule refl, simp)} 
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fun gen_abstract_equalities (dest : term > term * (term > term)) thm = 
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let 
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val thy = Thm.theory_of_thm thm 
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val prop = Thm.prop_of thm 
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val (t, mk_prop') = dest prop 
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val add_eqs = Term.fold_aterms 
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(fn t as Const (@{const_name HOL.eq}, _) => insert (op =) t  _ => I) 
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val eq_consts = rev (add_eqs t []) 
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val eqTs = map (snd o dest_Const) eq_consts 
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val used = Term.add_free_names prop [] 
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val names = map (K "") eqTs > Name.variant_list used 
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val frees = map Free (names ~~ eqTs) 
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val prems = map (HOLogic.mk_Trueprop o mk_is_equality) frees 
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val prop1 = mk_prop' (Term.subst_atomic (eq_consts ~~ frees) t) 
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val prop2 = fold Logic.all frees (Logic.list_implies (prems, prop1)) 
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val cprop = Thm.cterm_of thy prop2 
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val equal_thm = Raw_Simplifier.rewrite false [is_equality_lemma] cprop 
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fun forall_elim thm = Thm.forall_elim_vars (Thm.maxidx_of thm + 1) thm 
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in 
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forall_elim (thm COMP (equal_thm COMP @{thm equal_elim_rule2})) 
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end 
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handle TERM _ => thm 
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fun abstract_equalities_transfer thm = 
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let 
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fun dest prop = 
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let 
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val prems = Logic.strip_imp_prems prop 
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val concl = HOLogic.dest_Trueprop (Logic.strip_imp_concl prop) 
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val ((rel, x), y) = apfst Term.dest_comb (Term.dest_comb concl) 
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in 
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(rel, fn rel' => 
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Logic.list_implies (prems, HOLogic.mk_Trueprop (rel' $ x $ y))) 
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end 
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in 
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gen_abstract_equalities dest thm 
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end 
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fun abstract_equalities_relator_eq rel_eq_thm = 
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gen_abstract_equalities (fn x => (x, I)) 
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(rel_eq_thm RS @{thm is_equality_def [THEN iffD2]}) 
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(** Transfer proof method **) 
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val post_simps = 
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@{thms transfer_forall_eq [symmetric] 
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transfer_implies_eq [symmetric] transfer_bforall_unfold} 
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fun gen_frees_tac keepers ctxt = SUBGOAL (fn (t, i) => 
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let 
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val keepers = keepers @ get_known_frees ctxt 
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val vs = rev (Term.add_frees t []) 
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val vs' = filter_out (member (op =) keepers) vs 
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in 
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Induct.arbitrary_tac ctxt 0 vs' i 
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end) 
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fun mk_relT (T, U) = T > U > HOLogic.boolT 
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fun mk_Rel t = 
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let val T = fastype_of t 
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in Const (@{const_name Transfer.Rel}, T > T) $ t end 
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fun transfer_rule_of_terms ctxt tab t u = 
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let 
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val thy = Proof_Context.theory_of ctxt 
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(* precondition: T must consist of only TFrees and function space *) 
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fun rel (T as TFree (a, _)) U = 
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Free (the (AList.lookup (op =) tab a), mk_relT (T, U)) 
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 rel (T as Type ("fun", [T1, T2])) (U as Type ("fun", [U1, U2])) = 
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let 
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val r1 = rel T1 U1 
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val r2 = rel T2 U2 
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val rT = fastype_of r1 > fastype_of r2 > mk_relT (T, U) 
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in 
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Const (@{const_name fun_rel}, rT) $ r1 $ r2 
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end 
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 rel T U = raise TYPE ("rel", [T, U], []) 
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fun zip _ thms (Bound i) (Bound _) = (nth thms i, []) 
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 zip ctxt thms (Abs (x, T, t)) (Abs (y, U, u)) = 
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let 
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val ([x', y'], ctxt') = Variable.variant_fixes [x, y] ctxt 
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val prop = mk_Rel (rel T U) $ Free (x', T) $ Free (y', U) 
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val cprop = Thm.cterm_of thy (HOLogic.mk_Trueprop prop) 
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val thm0 = Thm.assume cprop 
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val (thm1, hyps) = zip ctxt' (thm0 :: thms) t u 
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val ((r1, x), y) = apfst Thm.dest_comb (Thm.dest_comb (Thm.dest_arg cprop)) 
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val r2 = Thm.dest_fun2 (Thm.dest_arg (cprop_of thm1)) 
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val (a1, (b1, _)) = apsnd dest_funcT (dest_funcT (ctyp_of_term r1)) 
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val (a2, (b2, _)) = apsnd dest_funcT (dest_funcT (ctyp_of_term r2)) 
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val tinsts = [SOME a1, SOME b1, SOME a2, SOME b2] 
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val insts = [SOME (Thm.dest_arg r1), SOME (Thm.dest_arg r2)] 
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val rule = Drule.instantiate' tinsts insts @{thm Rel_abs} 
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val thm2 = Thm.forall_intr x (Thm.forall_intr y (Thm.implies_intr cprop thm1)) 
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in 
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(thm2 COMP rule, hyps) 
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end 
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 zip ctxt thms (f $ t) (g $ u) = 
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let 
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val (thm1, hyps1) = zip ctxt thms f g 
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val (thm2, hyps2) = zip ctxt thms t u 
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in 
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(thm2 RS (thm1 RS @{thm Rel_app}), hyps1 @ hyps2) 
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end 
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 zip _ _ (t as Free (_, T)) u = 
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let 
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val U = fastype_of u 
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val prop = mk_Rel (rel T U) $ t $ u 
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val cprop = Thm.cterm_of thy (HOLogic.mk_Trueprop prop) 
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in 
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(Thm.assume cprop, [cprop]) 
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end 
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 zip _ _ t u = raise TERM ("zip_relterm", [t, u]) 
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val r = mk_Rel (rel (fastype_of t) (fastype_of u)) 
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val goal = HOLogic.mk_Trueprop (r $ t $ u) 
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val rename = Thm.trivial (cterm_of thy goal) 
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val (thm, hyps) = zip ctxt [] t u 
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in 
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Drule.implies_intr_list hyps (thm RS rename) 
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end 
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fun transfer_rule_of_term ctxt t = 
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let 
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val compound_rhs = get_compound_rhs ctxt 
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val is_rhs = not o null o Net.unify_term compound_rhs 
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fun dummy ctxt = 
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let 
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val (c, ctxt) = yield_singleton Variable.variant_fixes "a" ctxt 
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in 
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(Free (c, dummyT), ctxt) 
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end 
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(* create a lambda term of the same shape as the given term *) 
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fun skeleton (Bound i) ctxt = (Bound i, ctxt) 
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 skeleton (Abs (x, _, t)) ctxt = 
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let 
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val (t', ctxt) = skeleton t ctxt 
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in 
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(Abs (x, dummyT, t'), ctxt) 
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end 
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 skeleton (tu as (t $ u)) ctxt = 
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if is_rhs tu andalso not (Term.is_open tu) then dummy ctxt else 
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let 
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val (t', ctxt) = skeleton t ctxt 
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val (u', ctxt) = skeleton u ctxt 
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in 
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(t' $ u', ctxt) 
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end 
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 skeleton _ ctxt = dummy ctxt 
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val s = skeleton t ctxt > fst > Syntax.check_term ctxt > 
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map_types (map_type_tfree (fn (a, _) => TFree (a, HOLogic.typeS))) 
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val frees = map fst (Term.add_frees s []) 
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val tfrees = map fst (Term.add_tfrees s []) 
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fun prep a = "R" ^ Library.unprefix "'" a 
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val (rnames, ctxt') = Variable.variant_fixes (map prep tfrees) ctxt 
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val thm = transfer_rule_of_terms ctxt' (tfrees ~~ rnames) s t 
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in 
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Thm.generalize (tfrees, rnames @ frees) (Thm.maxidx_of thm + 1) thm 
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end 
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289 

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fun eq_tac eq_rules = TRY o REPEAT_ALL_NEW (resolve_tac eq_rules) THEN_ALL_NEW rtac @{thm is_equality_eq} 
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291 

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fun transfer_tac equiv ctxt i = 
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let 
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294 
val pre_simps = @{thms transfer_forall_eq transfer_implies_eq} 
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val start_rule = 
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if equiv then @{thm transfer_start} else @{thm transfer_start'} 
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val rules = get_transfer_raw ctxt 
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val eq_rules = get_relator_eq_raw ctxt 
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(* allow unsolved subgoals only for standard transfer method, not for transfer' *) 
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val end_tac = if equiv then K all_tac else K no_tac 
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val err_msg = "Transfer failed to convert goal to an objectlogic formula" 
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fun main_tac (t, i) = 
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rtac start_rule i THEN 
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(rtac (transfer_rule_of_term ctxt (HOLogic.dest_Trueprop t)) 
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THEN_ALL_NEW 
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(SOLVED' (REPEAT_ALL_NEW (resolve_tac rules) THEN_ALL_NEW (DETERM o eq_tac eq_rules)) 
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ORELSE' end_tac)) (i + 1) 
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handle TERM (_, ts) => raise TERM (err_msg, ts) 
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in 
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EVERY 

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[rewrite_goal_tac pre_simps i THEN 
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SUBGOAL main_tac i, 
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(* FIXME: rewrite_goal_tac does unwanted etacontraction *) 
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rewrite_goal_tac post_simps i, 
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rtac @{thm _} i] 
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end 
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fun transfer_prover_tac ctxt = SUBGOAL (fn (t, i) => 
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let 
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val rhs = (snd o Term.dest_comb o HOLogic.dest_Trueprop) t 
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val rule1 = transfer_rule_of_term ctxt rhs 
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val rules = get_transfer_raw ctxt 
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val eq_rules = get_relator_eq_raw ctxt 
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EVERY 

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[CONVERSION prep_conv i, 
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rtac @{thm transfer_prover_start} i, 
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(rtac rule1 THEN_ALL_NEW 
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(REPEAT_ALL_NEW (resolve_tac rules) THEN_ALL_NEW (DETERM o eq_tac eq_rules))) (i+1), 
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rtac @{thm refl} i] 
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end) 
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(** Methods and attributes **) 
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val free = Args.context  Args.term >> (fn (_, Free v) => v  (ctxt, t) => 
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error ("Bad free variable: " ^ Syntax.string_of_term ctxt t)) 
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val fixing = Scan.optional (Scan.lift (Args.$$$ "fixing"  Args.colon) 
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 Scan.repeat free) [] 
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fun transfer_method equiv : (Proof.context > Method.method) context_parser = 
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fixing >> (fn vs => fn ctxt => 
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SIMPLE_METHOD' (gen_frees_tac vs ctxt THEN' transfer_tac equiv ctxt)) 
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val transfer_prover_method : (Proof.context > Method.method) context_parser = 
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Scan.succeed (fn ctxt => SIMPLE_METHOD' (transfer_prover_tac ctxt)) 
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(* Attribute for transfer rules *) 
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val prep_rule = abstract_equalities_transfer o Conv.fconv_rule prep_conv 
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val transfer_add = 

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Thm.declaration_attribute (add_transfer_thm o prep_rule) 
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val transfer_del = 

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Thm.declaration_attribute (del_transfer_thm o prep_rule) 
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val transfer_attribute = 

359 
Attrib.add_del transfer_add transfer_del 

360 

361 
(* Theory setup *) 

362 

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val relator_eq_setup = 
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let 
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val name = @{binding relator_eq} 
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fun add_thm thm = Data.map (map_relator_eq (Item_Net.update thm)) 
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#> Data.map (map_relator_eq_raw (Item_Net.update (abstract_equalities_relator_eq thm))) 
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fun del_thm thm = Data.map (map_relator_eq (Item_Net.remove thm)) 
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#> Data.map (map_relator_eq_raw (Item_Net.remove (abstract_equalities_relator_eq thm))) 
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val add = Thm.declaration_attribute add_thm 
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val del = Thm.declaration_attribute del_thm 
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val text = "declaration of relator equality rule (used by transfer method)" 
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val content = Item_Net.content o #relator_eq o Data.get 
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in 
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Attrib.setup name (Attrib.add_del add del) text 
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#> Global_Theory.add_thms_dynamic (name, content) 
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end 
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val setup = 
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relator_eq_setup 
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#> Attrib.setup @{binding transfer_rule} transfer_attribute 
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"transfer rule for transfer method" 
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#> Global_Theory.add_thms_dynamic 
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(@{binding transfer_raw}, Item_Net.content o #transfer_raw o Data.get) 
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#> Global_Theory.add_thms_dynamic 
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(@{binding relator_eq_raw}, Item_Net.content o #relator_eq_raw o Data.get) 
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#> Method.setup @{binding transfer} (transfer_method true) 
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"generic theorem transfer method" 
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#> Method.setup @{binding transfer'} (transfer_method false) 
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"generic theorem transfer method" 
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#> Method.setup @{binding transfer_prover} transfer_prover_method 
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"for proving transfer rules" 
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394 
end 