src/HOL/simpdata.ML
author nipkow
Wed Dec 12 12:40:02 2001 +0100 (2001-12-12)
changeset 12475 18ba10cc782f
parent 12281 3bd113b8f7a6
child 12524 66eb843b1d35
permissions -rw-r--r--
Removed pointless backtracking from arith_tac
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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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(* legacy ML bindings *)
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val Eq_FalseI = thm "Eq_FalseI";
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val Eq_TrueI = thm "Eq_TrueI";
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val all_conj_distrib = thm "all_conj_distrib";
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val all_simps = thms "all_simps";
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val cases_simp = thm "cases_simp";
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val conj_assoc = thm "conj_assoc";
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val conj_comms = thms "conj_comms";
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val conj_commute = thm "conj_commute";
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val conj_cong = thm "conj_cong";
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val conj_disj_distribL = thm "conj_disj_distribL";
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val conj_disj_distribR = thm "conj_disj_distribR";
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val conj_left_commute = thm "conj_left_commute";
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val de_Morgan_conj = thm "de_Morgan_conj";
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val de_Morgan_disj = thm "de_Morgan_disj";
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val disj_assoc = thm "disj_assoc";
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val disj_comms = thms "disj_comms";
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val disj_commute = thm "disj_commute";
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val disj_cong = thm "disj_cong";
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val disj_conj_distribL = thm "disj_conj_distribL";
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val disj_conj_distribR = thm "disj_conj_distribR";
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val disj_left_commute = thm "disj_left_commute";
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val disj_not1 = thm "disj_not1";
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val disj_not2 = thm "disj_not2";
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val eq_ac = thms "eq_ac";
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val eq_assoc = thm "eq_assoc";
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val eq_commute = thm "eq_commute";
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val eq_left_commute = thm "eq_left_commute";
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val eq_sym_conv = thm "eq_sym_conv";
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val eta_contract_eq = thm "eta_contract_eq";
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val ex_disj_distrib = thm "ex_disj_distrib";
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val ex_simps = thms "ex_simps";
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val if_False = thm "if_False";
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val if_P = thm "if_P";
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val if_True = thm "if_True";
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val if_bool_eq_conj = thm "if_bool_eq_conj";
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val if_bool_eq_disj = thm "if_bool_eq_disj";
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val if_cancel = thm "if_cancel";
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val if_def2 = thm "if_def2";
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val if_eq_cancel = thm "if_eq_cancel";
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val if_not_P = thm "if_not_P";
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val if_splits = thms "if_splits";
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val iff_conv_conj_imp = thm "iff_conv_conj_imp";
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val imp_all = thm "imp_all";
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val imp_cong = thm "imp_cong";
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val imp_conjL = thm "imp_conjL";
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val imp_conjR = thm "imp_conjR";
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val imp_conv_disj = thm "imp_conv_disj";
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val imp_disj1 = thm "imp_disj1";
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val imp_disj2 = thm "imp_disj2";
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val imp_disjL = thm "imp_disjL";
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val imp_disj_not1 = thm "imp_disj_not1";
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val imp_disj_not2 = thm "imp_disj_not2";
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val imp_ex = thm "imp_ex";
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val meta_eq_to_obj_eq = thm "meta_eq_to_obj_eq";
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val neq_commute = thm "neq_commute";
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val not_all = thm "not_all";
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val not_ex = thm "not_ex";
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val not_iff = thm "not_iff";
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val not_imp = thm "not_imp";
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val not_not = thm "not_not";
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val rev_conj_cong = thm "rev_conj_cong";
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val simp_thms = thms "simp_thms";
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val split_if = thm "split_if";
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val split_if_asm = thm "split_if_asm";
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local
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val uncurry = prove_goal (the_context()) "P --> Q --> R ==> P & Q --> R"
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              (fn prems => [cut_facts_tac prems 1, Blast_tac 1]);
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val iff_allI = allI RS
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    prove_goal (the_context()) "!x. P x = Q x ==> (!x. P x) = (!x. Q x)"
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               (fn prems => [cut_facts_tac prems 1, Blast_tac 1])
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in
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(*** make simplification procedures for quantifier elimination ***)
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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 = eq_reflection
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  val iffI = iffI
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  val conjI= conjI
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  val conjE= conjE
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  val impI = impI
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  val mp   = mp
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  val uncurry = uncurry
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  val exI  = exI
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  val exE  = exE
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  val iff_allI = iff_allI
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end);
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end;
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local
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val ex_pattern = Thm.read_cterm (Theory.sign_of (the_context ()))
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    ("EX x. P(x) & Q(x)",HOLogic.boolT)
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val all_pattern = Thm.read_cterm (Theory.sign_of (the_context ()))
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    ("ALL x. P(x) --> Q(x)",HOLogic.boolT)
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in
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val defEX_regroup = mk_simproc "defined EX" [ex_pattern]
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      Quantifier1.rearrange_ex
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val defALL_regroup = mk_simproc "defined ALL" [all_pattern]
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      Quantifier1.rearrange_all
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end;
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(*** Case splitting ***)
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(*Make meta-equalities.  The operator below is Trueprop*)
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fun mk_meta_eq r = r RS eq_reflection;
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fun safe_mk_meta_eq r = mk_meta_eq r handle Thm.THM _ => r;
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fun mk_eq th = case concl_of th of
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        Const("==",_)$_$_       => th
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    |   _$(Const("op =",_)$_$_) => mk_meta_eq th
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    |   _$(Const("Not",_)$_)    => th RS Eq_FalseI
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    |   _                       => th RS Eq_TrueI;
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(* last 2 lines requires all formulae to be of the from Trueprop(.) *)
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fun mk_eq_True r =
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  Some (r RS meta_eq_to_obj_eq RS Eq_TrueI) handle Thm.THM _ => None;
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(*Congruence rules for = (instead of ==)*)
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fun mk_meta_cong rl =
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  standard(mk_meta_eq(replicate (nprems_of rl) meta_eq_to_obj_eq MRS rl))
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  handle THM _ =>
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  error("Premises and conclusion of congruence rules must be =-equalities");
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(* Elimination of True from asumptions: *)
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local fun rd s = read_cterm (sign_of (the_context())) (s, propT);
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in val True_implies_equals = standard' (equal_intr
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  (implies_intr_hyps (implies_elim (assume (rd "True ==> PROP P")) TrueI))
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  (implies_intr_hyps (implies_intr (rd "True") (assume (rd "PROP P")))));
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end;
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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 = meta_eq_to_obj_eq
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  val iffD           = iffD2
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  val disjE          = disjE
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  val conjE          = conjE
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  val exE            = exE
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  val contrapos      = contrapos_nn
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  val contrapos2     = contrapos_pp
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  val notnotD        = 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 split_asm_tac    = Splitter.split_asm_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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(*In general it seems wrong to add distributive laws by default: they
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  might cause exponential blow-up.  But imp_disjL has been in for a while
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  and cannot be removed without affecting existing proofs.  Moreover,
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  rewriting by "(P|Q --> R) = ((P-->R)&(Q-->R))" might be justified on the
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  grounds that it allows simplification of R in the two cases.*)
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val mksimps_pairs =
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  [("op -->", [mp]), ("op &", [conjunct1,conjunct2]),
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   ("All", [spec]), ("True", []), ("False", []),
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   ("If", [if_bool_eq_conj RS iffD1])];
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(* ###FIXME: move to Provers/simplifier.ML
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val mk_atomize:      (string * thm list) list -> thm -> thm list
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*)
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(* ###FIXME: move to Provers/simplifier.ML *)
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fun mk_atomize pairs =
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  let fun atoms th =
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        (case concl_of th of
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           Const("Trueprop",_) $ p =>
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             (case head_of p of
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                Const(a,_) =>
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                  (case assoc(pairs,a) of
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                     Some(rls) => flat (map atoms ([th] RL rls))
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                   | None => [th])
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              | _ => [th])
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         | _ => [th])
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  in atoms end;
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fun mksimps pairs =
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  (mapfilter (try mk_eq) o mk_atomize pairs o forall_elim_vars_safe);
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fun unsafe_solver_tac prems =
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  FIRST'[resolve_tac(reflexive_thm::TrueI::refl::prems), atac, etac FalseE];
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val unsafe_solver = mk_solver "HOL unsafe" unsafe_solver_tac;
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(*No premature instantiation of variables during simplification*)
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fun safe_solver_tac prems =
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  FIRST'[match_tac(reflexive_thm::TrueI::refl::prems),
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         eq_assume_tac, ematch_tac [FalseE]];
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val safe_solver = mk_solver "HOL safe" safe_solver_tac;
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val HOL_basic_ss =
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  empty_ss 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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val HOL_ss =
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    HOL_basic_ss addsimps
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     ([triv_forall_equality, (* prunes params *)
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       True_implies_equals, (* prune asms `True' *)
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       eta_contract_eq, (* prunes eta-expansions *)
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       if_True, if_False, if_cancel, if_eq_cancel,
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       imp_disjL, conj_assoc, disj_assoc,
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       de_Morgan_conj, de_Morgan_disj, imp_disj1, imp_disj2, not_imp,
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       disj_not1, not_all, not_ex, cases_simp,
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       thm "the_eq_trivial", the_sym_eq_trivial, thm "plus_ac0.zero", thm "plus_ac0_zero_right"]
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     @ ex_simps @ all_simps @ simp_thms)
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     addsimprocs [defALL_regroup,defEX_regroup]
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     addcongs [imp_cong]
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     addsplits [split_if];
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fun hol_simplify rews = Simplifier.full_simplify (HOL_basic_ss addsimps rews);
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(*Simplifies x assuming c and y assuming ~c*)
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val prems = Goalw [if_def]
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  "[| b=c; c ==> x=u; ~c ==> y=v |] ==> \
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\  (if b then x else y) = (if c then u else v)";
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by (asm_simp_tac (HOL_ss addsimps prems) 1);
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qed "if_cong";
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(*Prevents simplification of x and y: faster and allows the execution
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  of functional programs. NOW THE DEFAULT.*)
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Goal "b=c ==> (if b then x else y) = (if c then x else y)";
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by (etac arg_cong 1);
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qed "if_weak_cong";
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(*Prevents simplification of t: much faster*)
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Goal "a = b ==> (let x=a in t(x)) = (let x=b in t(x))";
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by (etac arg_cong 1);
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qed "let_weak_cong";
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Goal "f(if c then x else y) = (if c then f x else f y)";
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by (simp_tac (HOL_ss setloop (split_tac [split_if])) 1);
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qed "if_distrib";
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(*For expand_case_tac*)
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val prems = Goal "[| P ==> Q(True); ~P ==> Q(False) |] ==> Q(P)";
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by (case_tac "P" 1);
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by (ALLGOALS (asm_simp_tac (HOL_ss addsimps prems)));
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qed "expand_case";
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(*Used in Auth proofs.  Typically P contains Vars that become instantiated
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  during unification.*)
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fun expand_case_tac P i =
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    res_inst_tac [("P",P)] expand_case i THEN
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    Simp_tac (i+1) THEN
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    Simp_tac i;
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(*This lemma restricts the effect of the rewrite rule u=v to the left-hand
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  side of an equality.  Used in {Integ,Real}/simproc.ML*)
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Goal "x=y ==> (x=z) = (y=z)";
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by (asm_simp_tac HOL_ss 1);
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qed "restrict_to_left";
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(* default simpset *)
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val simpsetup =
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  [fn thy => (simpset_ref_of thy := HOL_ss addcongs [if_weak_cong]; thy)];
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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 = iffD1 val iffD2 = iffD2 val notE = notE
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  val cla_make_elim = cla_make_elim);
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open Clasimp;
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val HOL_css = (HOL_cs, HOL_ss);
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(*** A general 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 top-level &, | or EX
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*)
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fun refute_tac test prep_tac ref_tac =
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  let val nnf_simps =
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        [imp_conv_disj,iff_conv_conj_imp,de_Morgan_disj,de_Morgan_conj,
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         not_all,not_ex,not_not];
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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)
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                 addsimps nnf_simps;
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      val prem_nnf_tac = full_simp_tac nnf_simpset;
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      val refute_prems_tac =
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        REPEAT_DETERM
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              (eresolve_tac [conjE, exE] 1 ORELSE
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               filter_prems_tac test 1 ORELSE
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               etac disjE 1) THEN
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        ((etac notE 1 THEN eq_assume_tac 1) ORELSE
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         ref_tac 1);
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  in EVERY'[TRY o filter_prems_tac test,
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            REPEAT_DETERM o etac rev_mp, prep_tac, rtac ccontr, prem_nnf_tac,
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            SELECT_GOAL (DEPTH_SOLVE refute_prems_tac)]
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  end;