author  haftmann 
Sun, 06 May 2007 21:49:23 +0200  
changeset 22838  466599ecf610 
parent 22147  f4ed4d940d44 
child 23199  42004f6d908b 
permissions  rwrr 
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(* Title: HOL/simpdata.ML 
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ID: $Id$ 

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Author: Tobias Nipkow 

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Copyright 1991 University of Cambridge 

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Instantiation of the generic simplifier for HOL. 

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*) 

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(** tools setup **) 

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structure Quantifier1 = Quantifier1Fun 

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(struct 

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(*abstract syntax*) 

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fun dest_eq ((c as Const("op =",_)) $ s $ t) = SOME (c, s, t) 

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 dest_eq _ = NONE; 

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fun dest_conj ((c as Const("op &",_)) $ s $ t) = SOME (c, s, t) 

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 dest_conj _ = NONE; 

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fun dest_imp ((c as Const("op >",_)) $ s $ t) = SOME (c, s, t) 

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 dest_imp _ = NONE; 

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val conj = HOLogic.conj 

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val imp = HOLogic.imp 

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(*rules*) 

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val iff_reflection = @{thm eq_reflection} 
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val iffI = @{thm iffI} 

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val iff_trans = @{thm trans} 

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val conjI= @{thm conjI} 

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val conjE= @{thm conjE} 

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val impI = @{thm impI} 

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val mp = @{thm mp} 

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val uncurry = @{thm uncurry} 

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val exI = @{thm exI} 

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val exE = @{thm exE} 

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val iff_allI = @{thm iff_allI} 

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val iff_exI = @{thm iff_exI} 

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val all_comm = @{thm all_comm} 

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val ex_comm = @{thm ex_comm} 

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end); 
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structure Simpdata = 
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struct 
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fun mk_meta_eq r = r RS @{thm eq_reflection}; 
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fun safe_mk_meta_eq r = mk_meta_eq r handle Thm.THM _ => r; 
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22147  45 
fun mk_eq th = case concl_of th 
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(*expects Trueprop if not == *) 
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of Const ("==",_) $ _ $ _ => th 
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 _ $ (Const ("op =", _) $ _ $ _) => mk_meta_eq th 

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 _ $ (Const ("Not", _) $ _) => th RS @{thm Eq_FalseI} 
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 _ => th RS @{thm Eq_TrueI} 

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fun mk_eq_True r = 
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SOME (r RS @{thm meta_eq_to_obj_eq} RS @{thm Eq_TrueI}) handle Thm.THM _ => NONE; 

21163  54 

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(* Produce theorems of the form 

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(P1 =simp=> ... =simp=> Pn => x == y) ==> (P1 =simp=> ... =simp=> Pn => x = y) 

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*) 

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22147  59 
fun lift_meta_eq_to_obj_eq i st = 
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let 
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fun count_imp (Const ("HOL.simp_implies", _) $ P $ Q) = 1 + count_imp Q 

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 count_imp _ = 0; 

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val j = count_imp (Logic.strip_assums_concl (List.nth (prems_of st, i  1))) 

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in if j = 0 then @{thm meta_eq_to_obj_eq} 
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else 
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let 

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val Ps = map (fn k => Free ("P" ^ string_of_int k, propT)) (1 upto j); 

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fun mk_simp_implies Q = foldr (fn (R, S) => 

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Const ("HOL.simp_implies", propT > propT > propT) $ R $ S) Q Ps 

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val aT = TFree ("'a", HOLogic.typeS); 

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val x = Free ("x", aT); 

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val y = Free ("y", aT) 

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in Goal.prove_global (Thm.theory_of_thm st) [] 

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[mk_simp_implies (Logic.mk_equals (x, y))] 

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(mk_simp_implies (HOLogic.mk_Trueprop (HOLogic.mk_eq (x, y)))) 

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(fn prems => EVERY 

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[rewrite_goals_tac @{thms simp_implies_def}, 
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REPEAT (ares_tac (@{thm meta_eq_to_obj_eq} :: 

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map (rewrite_rule @{thms simp_implies_def}) prems) 1)]) 

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end 
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end; 

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(*Congruence rules for = (instead of ==)*) 

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fun mk_meta_cong rl = zero_var_indexes 

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(let val rl' = Seq.hd (TRYALL (fn i => fn st => 

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rtac (lift_meta_eq_to_obj_eq i st) i st) rl) 

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in mk_meta_eq rl' handle THM _ => 

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if can Logic.dest_equals (concl_of rl') then rl' 

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else error "Conclusion of congruence rules must be =equality" 

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end); 

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fun mk_atomize pairs = 

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let 

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fun atoms thm = 
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let 
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fun res th = map (fn rl => th RS rl); (*exception THM*) 
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fun res_fixed rls = 
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if Thm.maxidx_of (Thm.adjust_maxidx_thm ~1 thm) = ~1 then res thm rls 
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else Variable.trade (K (fn [thm'] => res thm' rls)) (Variable.thm_context thm) [thm]; 
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in 
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case concl_of thm 
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of Const ("Trueprop", _) $ p => (case head_of p 
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of Const (a, _) => (case AList.lookup (op =) pairs a 
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of SOME rls => (maps atoms (res_fixed rls) handle THM _ => [thm]) 
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 NONE => [thm]) 
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 _ => [thm]) 
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 _ => [thm] 
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end; 
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in atoms end; 
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fun mksimps pairs = 

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map_filter (try mk_eq) o mk_atomize pairs o gen_all; 
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fun unsafe_solver_tac prems = 
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(fn i => REPEAT_DETERM (match_tac @{thms simp_impliesI} i)) THEN' 

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FIRST' [resolve_tac (reflexive_thm :: @{thm TrueI} :: @{thm refl} :: prems), atac, 

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etac @{thm FalseE}]; 

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val unsafe_solver = mk_solver "HOL unsafe" unsafe_solver_tac; 
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22838  121 

21163  122 
(*No premature instantiation of variables during simplification*) 
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fun safe_solver_tac prems = 
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(fn i => REPEAT_DETERM (match_tac @{thms simp_impliesI} i)) THEN' 

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FIRST' [match_tac (reflexive_thm :: @{thm TrueI} :: @{thm refl} :: prems), 

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eq_assume_tac, ematch_tac @{thms FalseE}]; 

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21163  128 
val safe_solver = mk_solver "HOL safe" safe_solver_tac; 
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structure SplitterData = 

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struct 

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structure Simplifier = Simplifier 

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val mk_eq = mk_eq 
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val meta_eq_to_iff = @{thm meta_eq_to_obj_eq} 
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val iffD = @{thm iffD2} 

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val disjE = @{thm disjE} 

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val conjE = @{thm conjE} 

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val exE = @{thm exE} 

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val contrapos = @{thm contrapos_nn} 

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val contrapos2 = @{thm contrapos_pp} 

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val notnotD = @{thm notnotD} 

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end; 
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structure Splitter = SplitterFun(SplitterData); 

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val split_tac = Splitter.split_tac; 
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val split_inside_tac = Splitter.split_inside_tac; 

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val op addsplits = Splitter.addsplits; 

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val op delsplits = Splitter.delsplits; 

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val Addsplits = Splitter.Addsplits; 

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val Delsplits = Splitter.Delsplits; 

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(* integration of simplifier with classical reasoner *) 
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structure Clasimp = ClasimpFun 

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(structure Simplifier = Simplifier and Splitter = Splitter 

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and Classical = Classical and Blast = Blast 

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val iffD1 = @{thm iffD1} val iffD2 = @{thm iffD2} val notE = @{thm notE}); 
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open Clasimp; 
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val _ = ML_Context.value_antiq "clasimpset" 
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(Scan.succeed ("clasimpset", "Clasimp.local_clasimpset_of (ML_Context.the_local_context ())")); 

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val mksimps_pairs = 
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[("op >", [@{thm mp}]), ("op &", [@{thm conjunct1}, @{thm conjunct2}]), 
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("All", [@{thm spec}]), ("True", []), ("False", []), 

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("HOL.If", [@{thm if_bool_eq_conj} RS @{thm iffD1}])]; 

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val HOL_basic_ss = 
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Simplifier.theory_context @{theory} empty_ss 
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setsubgoaler asm_simp_tac 
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setSSolver safe_solver 

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setSolver unsafe_solver 

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setmksimps (mksimps mksimps_pairs) 

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setmkeqTrue mk_eq_True 

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setmkcong mk_meta_cong; 

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fun hol_simplify rews = Simplifier.full_simplify (HOL_basic_ss addsimps rews); 
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fun unfold_tac ths = 

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let val ss0 = Simplifier.clear_ss HOL_basic_ss addsimps ths 
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in fn ss => ALLGOALS (full_simp_tac (Simplifier.inherit_context ss ss0)) end; 
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(** simprocs **) 
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(* simproc for proving "(y = x) == False" from premise "~(x = y)" *) 

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val use_neq_simproc = ref true; 

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local 

22147  195 
val neq_to_EQ_False = @{thm not_sym} RS @{thm Eq_FalseI}; 
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fun neq_prover sg ss (eq $ lhs $ rhs) = 
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let 

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fun test thm = (case #prop (rep_thm thm) of 

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_ $ (Not $ (eq' $ l' $ r')) => 

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Not = HOLogic.Not andalso eq' = eq andalso 

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r' aconv lhs andalso l' aconv rhs 

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 _ => false) 

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in if !use_neq_simproc then case find_first test (prems_of_ss ss) 

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of NONE => NONE 

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 SOME thm => SOME (thm RS neq_to_EQ_False) 

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else NONE 

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end 

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in 

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val neq_simproc = Simplifier.simproc @{theory} "neq_simproc" ["x = y"] neq_prover; 
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end; 
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(* simproc for Let *) 

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val use_let_simproc = ref true; 

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local 

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val (f_Let_unfold, x_Let_unfold) = 

22147  221 
let val [(_$(f$x)$_)] = prems_of @{thm Let_unfold} 
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in (cterm_of @{theory} f, cterm_of @{theory} x) end 

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val (f_Let_folded, x_Let_folded) = 
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let val [(_$(f$x)$_)] = prems_of @{thm Let_folded} 
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in (cterm_of @{theory} f, cterm_of @{theory} x) end; 

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val g_Let_folded = 
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let val [(_$_$(g$_))] = prems_of @{thm Let_folded} in cterm_of @{theory} g end; 
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in 
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val let_simproc = 

22147  231 
Simplifier.simproc @{theory} "let_simp" ["Let x f"] 
22838  232 
(fn thy => fn ss => fn t => 
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let val ctxt = Simplifier.the_context ss; 
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val ([t'], ctxt') = Variable.import_terms false [t] ctxt; 

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in Option.map (hd o Variable.export ctxt' ctxt o single) 

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(case t' of (Const ("Let",_)$x$f) => (* x and f are already in normal form *) 

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if not (!use_let_simproc) then NONE 

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else if is_Free x orelse is_Bound x orelse is_Const x 

22147  239 
then SOME @{thm Let_def} 
21163  240 
else 
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let 

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val n = case f of (Abs (x,_,_)) => x  _ => "x"; 

22838  243 
val cx = cterm_of thy x; 
21163  244 
val {T=xT,...} = rep_cterm cx; 
22838  245 
val cf = cterm_of thy f; 
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val fx_g = Simplifier.rewrite ss (Thm.capply cf cx); 
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val (_$_$g) = prop_of fx_g; 

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val g' = abstract_over (x,g); 

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in (if (g aconv g') 

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then 

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let 

22147  252 
val rl = 
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cterm_instantiate [(f_Let_unfold,cf),(x_Let_unfold,cx)] @{thm Let_unfold}; 

21163  254 
in SOME (rl OF [fx_g]) end 
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else if Term.betapply (f,x) aconv g then NONE (*avoid identity conversion*) 

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else let 

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val abs_g'= Abs (n,xT,g'); 

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val g'x = abs_g'$x; 

22838  259 
val g_g'x = symmetric (beta_conversion false (cterm_of thy g'x)); 
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val rl = cterm_instantiate 
22838  261 
[(f_Let_folded,cterm_of thy f),(x_Let_folded,cx), 
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(g_Let_folded,cterm_of thy abs_g')] 

22147  263 
@{thm Let_folded}; 
21163  264 
in SOME (rl OF [transitive fx_g g_g'x]) 
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end) 

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end 

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 _ => NONE) 

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end) 

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end; 
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(* generic refutation procedure *) 

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(* parameters: 

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test: term > bool 

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tests if a term is at all relevant to the refutation proof; 

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if not, then it can be discarded. Can improve performance, 

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esp. if disjunctions can be discarded (no case distinction needed!). 

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prep_tac: int > tactic 

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A preparation tactic to be applied to the goal once all relevant premises 

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have been moved to the conclusion. 

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ref_tac: int > tactic 

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the actual refutation tactic. Should be able to deal with goals 

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[ A1; ...; An ] ==> False 

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where the Ai are atomic, i.e. no toplevel &,  or EX 

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*) 

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local 

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val nnf_simpset = 

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empty_ss setmkeqTrue mk_eq_True 

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setmksimps (mksimps mksimps_pairs) 

22147  296 
addsimps [@{thm imp_conv_disj}, @{thm iff_conv_conj_imp}, @{thm de_Morgan_disj}, 
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@{thm de_Morgan_conj}, @{thm not_all}, @{thm not_ex}, @{thm not_not}]; 

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fun prem_nnf_tac i st = 
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full_simp_tac (Simplifier.theory_context (Thm.theory_of_thm st) nnf_simpset) i st; 

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in 

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fun refute_tac test prep_tac ref_tac = 

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let val refute_prems_tac = 

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REPEAT_DETERM 

22147  304 
(eresolve_tac [@{thm conjE}, @{thm exE}] 1 ORELSE 
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filter_prems_tac test 1 ORELSE 
22147  306 
etac @{thm disjE} 1) THEN 
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((etac @{thm notE} 1 THEN eq_assume_tac 1) ORELSE 

21163  308 
ref_tac 1); 
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in EVERY'[TRY o filter_prems_tac test, 

22147  310 
REPEAT_DETERM o etac @{thm rev_mp}, prep_tac, rtac @{thm ccontr}, prem_nnf_tac, 
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SELECT_GOAL (DEPTH_SOLVE refute_prems_tac)] 
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end; 

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end; 
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val defALL_regroup = 

22147  316 
Simplifier.simproc @{theory} 
21163  317 
"defined ALL" ["ALL x. P x"] Quantifier1.rearrange_all; 
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val defEX_regroup = 

22147  320 
Simplifier.simproc @{theory} 
21163  321 
"defined EX" ["EX x. P x"] Quantifier1.rearrange_ex; 
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21674  324 
val simpset_simprocs = HOL_basic_ss 
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addsimprocs [defALL_regroup, defEX_regroup, neq_simproc, let_simproc] 
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end; 
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structure Splitter = Simpdata.Splitter; 

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structure Clasimp = Simpdata.Clasimp; 