# File simpdata.ML

    (*  Title:      HOL/simpdata.ML
        ID:         $Id: simpdata.ML,v 1.167 2005/09/12 16:22:58 haftmann Exp $
        Author:     Tobias Nipkow
        Copyright   1991  University of Cambridge

    Instantiation of the generic simplifier for HOL.
    *)

    (* legacy ML bindings *)

    val Eq_FalseI = thm "Eq_FalseI";
    val Eq_TrueI = thm "Eq_TrueI";
    val all_conj_distrib = thm "all_conj_distrib";
    val all_simps = thms "all_simps";
    val cases_simp = thm "cases_simp";
    val conj_assoc = thm "conj_assoc";
    val conj_comms = thms "conj_comms";
    val conj_commute = thm "conj_commute";
    val conj_cong = thm "conj_cong";
    val conj_disj_distribL = thm "conj_disj_distribL";
    val conj_disj_distribR = thm "conj_disj_distribR";
    val conj_left_commute = thm "conj_left_commute";
    val de_Morgan_conj = thm "de_Morgan_conj";
    val de_Morgan_disj = thm "de_Morgan_disj";
    val disj_assoc = thm "disj_assoc";
    val disj_comms = thms "disj_comms";
    val disj_commute = thm "disj_commute";
    val disj_cong = thm "disj_cong";
    val disj_conj_distribL = thm "disj_conj_distribL";
    val disj_conj_distribR = thm "disj_conj_distribR";
    val disj_left_commute = thm "disj_left_commute";
    val disj_not1 = thm "disj_not1";
    val disj_not2 = thm "disj_not2";
    val eq_ac = thms "eq_ac";
    val eq_assoc = thm "eq_assoc";
    val eq_commute = thm "eq_commute";
    val eq_left_commute = thm "eq_left_commute";
    val eq_sym_conv = thm "eq_sym_conv";
    val eta_contract_eq = thm "eta_contract_eq";
    val ex_disj_distrib = thm "ex_disj_distrib";
    val ex_simps = thms "ex_simps";
    val if_False = thm "if_False";
    val if_P = thm "if_P";
    val if_True = thm "if_True";
    val if_bool_eq_conj = thm "if_bool_eq_conj";
    val if_bool_eq_disj = thm "if_bool_eq_disj";
    val if_cancel = thm "if_cancel";
    val if_def2 = thm "if_def2";
    val if_eq_cancel = thm "if_eq_cancel";
    val if_not_P = thm "if_not_P";
    val if_splits = thms "if_splits";
    val iff_conv_conj_imp = thm "iff_conv_conj_imp";
    val imp_all = thm "imp_all";
    val imp_cong = thm "imp_cong";
    val imp_conjL = thm "imp_conjL";
    val imp_conjR = thm "imp_conjR";
    val imp_conv_disj = thm "imp_conv_disj";
    val imp_disj1 = thm "imp_disj1";
    val imp_disj2 = thm "imp_disj2";
    val imp_disjL = thm "imp_disjL";
    val imp_disj_not1 = thm "imp_disj_not1";
    val imp_disj_not2 = thm "imp_disj_not2";
    val imp_ex = thm "imp_ex";
    val meta_eq_to_obj_eq = thm "meta_eq_to_obj_eq";
    val neq_commute = thm "neq_commute";
    val not_all = thm "not_all";
    val not_ex = thm "not_ex";
    val not_iff = thm "not_iff";
    val not_imp = thm "not_imp";
    val not_not = thm "not_not";
    val rev_conj_cong = thm "rev_conj_cong";
    val simp_impliesE = thm "simp_impliesI";
    val simp_impliesI = thm "simp_impliesI";
    val simp_implies_cong = thm "simp_implies_cong";
    val simp_implies_def = thm "simp_implies_def";
    val simp_thms = thms "simp_thms";
    val split_if = thm "split_if";
    val split_if_asm = thm "split_if_asm";
    val atomize_not = thm"atomize_not";

    local
    val uncurry = prove_goal (the_context()) "P --> Q --> R ==> P & Q --> R"
                  (fn prems => [cut_facts_tac prems 1, Blast_tac 1]);

    val iff_allI = allI RS
        prove_goal (the_context()) "!x. P x = Q x ==> (!x. P x) = (!x. Q x)"
                   (fn prems => [cut_facts_tac prems 1, Blast_tac 1])
    val iff_exI = allI RS
        prove_goal (the_context()) "!x. P x = Q x ==> (? x. P x) = (? x. Q x)"
                   (fn prems => [cut_facts_tac prems 1, Blast_tac 1])

    val all_comm = prove_goal (the_context()) "(!x y. P x y) = (!y x. P x y)"
                   (fn _ => [Blast_tac 1])
    val ex_comm = prove_goal (the_context()) "(? x y. P x y) = (? y x. P x y)"
                   (fn _ => [Blast_tac 1])
    in

    (*** make simplification procedures for quantifier elimination ***)

    structure Quantifier1 = Quantifier1Fun
    (struct
      (*abstract syntax*)
      fun dest_eq((c as Const("op =",_)) $ s $ t) = SOME(c,s,t)
        | dest_eq _ = NONE;
      fun dest_conj((c as Const("op &",_)) $ s $ t) = SOME(c,s,t)
        | dest_conj _ = NONE;
      fun dest_imp((c as Const("op -->",_)) $ s $ t) = SOME(c,s,t)
        | dest_imp _ = NONE;
      val conj = HOLogic.conj
      val imp  = HOLogic.imp
      (*rules*)
      val iff_reflection = eq_reflection
      val iffI = iffI
      val iff_trans = trans
      val conjI= conjI
      val conjE= conjE
      val impI = impI
      val mp   = mp
      val uncurry = uncurry
      val exI  = exI
      val exE  = exE
      val iff_allI = iff_allI
      val iff_exI = iff_exI
      val all_comm = all_comm
      val ex_comm = ex_comm
    end);

    end;

    val defEX_regroup =
      Simplifier.simproc (Theory.sign_of (the_context ()))
        "defined EX" ["EX x. P x"] Quantifier1.rearrange_ex;

    val defALL_regroup =
      Simplifier.simproc (Theory.sign_of (the_context ()))
        "defined ALL" ["ALL x. P x"] Quantifier1.rearrange_all;

    (*** Simproc for Let ***)

    val use_let_simproc = ref true;

    local
    val Let_folded = thm "Let_folded";
    val Let_unfold = thm "Let_unfold";

    val f_Let_unfold =
          let val [(_$(f$_)$_)] = prems_of Let_unfold in cterm_of (sign_of (the_context ())) f end
    val f_Let_folded =
          let val [(_$(f$_)$_)] = prems_of Let_folded in cterm_of (sign_of (the_context ())) f end;
    val g_Let_folded =
          let val [(_$_$(g$_))] = prems_of Let_folded in cterm_of (sign_of (the_context ())) g end;
    in
    val let_simproc =
      Simplifier.simproc (Theory.sign_of (the_context ())) "let_simp" ["Let x f"]
       (fn sg => fn ss => fn t =>
          (case t of (Const ("Let",_)$x$f) => (* x and f are already in normal form *)
             if not (!use_let_simproc) then NONE
             else if is_Free x orelse is_Bound x orelse is_Const x
             then SOME Let_def
             else
              let
                 val n = case f of (Abs (x,_,_)) => x | _ => "x";
                 val cx = cterm_of sg x;
                 val {T=xT,...} = rep_cterm cx;
                 val cf = cterm_of sg f;
                 val fx_g = Simplifier.rewrite ss (Thm.capply cf cx);
                 val (_$_$g) = prop_of fx_g;
                 val g' = abstract_over (x,g);
               in (if (g aconv g')
                   then
                      let
                        val rl = cterm_instantiate [(f_Let_unfold,cf)] Let_unfold;
                      in SOME (rl OF [fx_g]) end
                   else if betapply (f,x) aconv g then NONE (* avoid identity conversion *)
                   else let
                         val abs_g'= Abs (n,xT,g');
                         val g'x = abs_g'$x;
                         val g_g'x = symmetric (beta_conversion false (cterm_of sg g'x));
                         val rl = cterm_instantiate
                                   [(f_Let_folded,cterm_of sg f),
                                    (g_Let_folded,cterm_of sg abs_g')]
                                   Let_folded;
                       in SOME (rl OF [transitive fx_g g_g'x]) end)
               end
            | _ => NONE))
    end

    (*** Case splitting ***)

    (*Make meta-equalities.  The operator below is Trueprop*)

    fun mk_meta_eq r = r RS eq_reflection;
    fun safe_mk_meta_eq r = mk_meta_eq r handle Thm.THM _ => r;

    fun mk_eq th = case concl_of th of
            Const("==",_)$_$_       => th
        |   _$(Const("op =",_)$_$_) => mk_meta_eq th
        |   _$(Const("Not",_)$_)    => th RS Eq_FalseI
        |   _                       => th RS Eq_TrueI;
    (* Expects Trueprop(.) if not == *)

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

    (* Produce theorems of the form
      (P1 =simp=> ... =simp=> Pn => x == y) ==> (P1 =simp=> ... =simp=> Pn => x = y)
    *)
    fun lift_meta_eq_to_obj_eq i st =
      let
        val {sign, ...} = rep_thm st;
        fun count_imp (Const ("HOL.simp_implies", _) $ P $ Q) = 1 + count_imp Q
          | count_imp _ = 0;
        val j = count_imp (Logic.strip_assums_concl (List.nth (prems_of st, i - 1)))
      in if j = 0 then meta_eq_to_obj_eq
        else
          let
            val Ps = map (fn k => Free ("P" ^ string_of_int k, propT)) (1 upto j);
            fun mk_simp_implies Q = foldr (fn (R, S) =>
              Const ("HOL.simp_implies", propT --> propT --> propT) $ R $ S) Q Ps
            val aT = TFree ("'a", HOLogic.typeS);
            val x = Free ("x", aT);
            val y = Free ("y", aT)
          in prove_goalw_cterm [simp_implies_def]
            (cterm_of sign (Logic.mk_implies
              (mk_simp_implies (Logic.mk_equals (x, y)),
               mk_simp_implies (HOLogic.mk_Trueprop (HOLogic.mk_eq (x, y))))))
            (fn hyps => [REPEAT (ares_tac (meta_eq_to_obj_eq :: hyps) 1)])
          end
      end;

    (*Congruence rules for = (instead of ==)*)
    fun mk_meta_cong rl = zero_var_indexes
      (let val rl' = Seq.hd (TRYALL (fn i => fn st =>
         rtac (lift_meta_eq_to_obj_eq i st) i st) rl)
       in mk_meta_eq rl' handle THM _ =>
         if Logic.is_equals (concl_of rl') then rl'
         else error "Conclusion of congruence rules must be =-equality"
       end);

    (* Elimination of True from asumptions: *)

    local fun rd s = read_cterm (sign_of (the_context())) (s, propT);
    in val True_implies_equals = standard' (equal_intr
      (implies_intr_hyps (implies_elim (assume (rd "True ==> PROP P")) TrueI))
      (implies_intr_hyps (implies_intr (rd "True") (assume (rd "PROP P")))));
    end;

    structure SplitterData =
      struct
      structure Simplifier = Simplifier
      val mk_eq          = mk_eq
      val meta_eq_to_iff = meta_eq_to_obj_eq
      val iffD           = iffD2
      val disjE          = disjE
      val conjE          = conjE
      val exE            = exE
      val contrapos      = contrapos_nn
      val contrapos2     = contrapos_pp
      val notnotD        = notnotD
      end;

    structure Splitter = SplitterFun(SplitterData);

    val split_tac        = Splitter.split_tac;
    val split_inside_tac = Splitter.split_inside_tac;
    val split_asm_tac    = Splitter.split_asm_tac;
    val op addsplits     = Splitter.addsplits;
    val op delsplits     = Splitter.delsplits;
    val Addsplits        = Splitter.Addsplits;
    val Delsplits        = Splitter.Delsplits;

    val mksimps_pairs =
      [("op -->", [mp]), ("op &", [conjunct1,conjunct2]),
       ("All", [spec]), ("True", []), ("False", []),
       ("HOL.If", [if_bool_eq_conj RS iffD1])];

    (*
    val mk_atomize:      (string * thm list) list -> thm -> thm list
    looks too specific to move it somewhere else
    *)
    fun mk_atomize pairs =
      let fun atoms th =
            (case concl_of th of
               Const("Trueprop",_) $ p =>
                 (case head_of p of
                    Const(a,_) =>
                      (case AList.lookup (op =) pairs a of
                         SOME(rls) => List.concat (map atoms ([th] RL rls))
                       | NONE => [th])
                  | _ => [th])
             | _ => [th])
      in atoms end;

    fun mksimps pairs =
      (List.mapPartial (try mk_eq) o mk_atomize pairs o gen_all);

    fun unsafe_solver_tac prems =
      (fn i => REPEAT_DETERM (match_tac [simp_impliesI] i)) THEN'
      FIRST'[resolve_tac(reflexive_thm::TrueI::refl::prems), atac, etac FalseE];
    val unsafe_solver = mk_solver "HOL unsafe" unsafe_solver_tac;

    (*No premature instantiation of variables during simplification*)
    fun safe_solver_tac prems =
      (fn i => REPEAT_DETERM (match_tac [simp_impliesI] i)) THEN'
      FIRST'[match_tac(reflexive_thm::TrueI::refl::prems),
             eq_assume_tac, ematch_tac [FalseE]];
    val safe_solver = mk_solver "HOL safe" safe_solver_tac;

    val HOL_basic_ss =
      empty_ss setsubgoaler asm_simp_tac
        setSSolver safe_solver
        setSolver unsafe_solver
        setmksimps (mksimps mksimps_pairs)
        setmkeqTrue mk_eq_True
        setmkcong mk_meta_cong;

    fun unfold_tac ss ths =
      ALLGOALS (full_simp_tac
        (Simplifier.inherit_bounds ss (Simplifier.clear_ss HOL_basic_ss) addsimps ths));

    (*In general it seems wrong to add distributive laws by default: they
      might cause exponential blow-up.  But imp_disjL has been in for a while
      and cannot be removed without affecting existing proofs.  Moreover,
      rewriting by "(P|Q --> R) = ((P-->R)&(Q-->R))" might be justified on the
      grounds that it allows simplification of R in the two cases.*)

    val HOL_ss =
        HOL_basic_ss addsimps
         ([triv_forall_equality, (* prunes params *)
           True_implies_equals, (* prune asms `True' *)
           eta_contract_eq, (* prunes eta-expansions *)
           if_True, if_False, if_cancel, if_eq_cancel,
           imp_disjL, conj_assoc, disj_assoc,
           de_Morgan_conj, de_Morgan_disj, imp_disj1, imp_disj2, not_imp,
           disj_not1, not_all, not_ex, cases_simp,
           thm "the_eq_trivial", the_sym_eq_trivial]
         @ ex_simps @ all_simps @ simp_thms)
         addsimprocs [defALL_regroup,defEX_regroup,let_simproc]
         addcongs [imp_cong, simp_implies_cong]
         addsplits [split_if];

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

    (*Simplifies x assuming c and y assuming ~c*)
    val prems = Goalw [if_def]
      "[| b=c; c ==> x=u; ~c ==> y=v |] ==> \
    \  (if b then x else y) = (if c then u else v)";
    by (asm_simp_tac (HOL_ss addsimps prems) 1);
    qed "if_cong";

    (*Prevents simplification of x and y: faster and allows the execution
      of functional programs. NOW THE DEFAULT.*)
    Goal "b=c ==> (if b then x else y) = (if c then x else y)";
    by (etac arg_cong 1);
    qed "if_weak_cong";

    (*Prevents simplification of t: much faster*)
    Goal "a = b ==> (let x=a in t(x)) = (let x=b in t(x))";
    by (etac arg_cong 1);
    qed "let_weak_cong";

    (*To tidy up the result of a simproc.  Only the RHS will be simplified.*)
    Goal "u = u' ==> (t==u) == (t==u')";
    by (asm_simp_tac HOL_ss 1);
    qed "eq_cong2";

    Goal "f(if c then x else y) = (if c then f x else f y)";
    by (simp_tac (HOL_ss setloop (split_tac [split_if])) 1);
    qed "if_distrib";

    (*For expand_case_tac*)
    val prems = Goal "[| P ==> Q(True); ~P ==> Q(False) |] ==> Q(P)";
    by (case_tac "P" 1);
    by (ALLGOALS (asm_simp_tac (HOL_ss addsimps prems)));
    qed "expand_case";

    (*Used in Auth proofs.  Typically P contains Vars that become instantiated
      during unification.*)
    fun expand_case_tac P i =
        res_inst_tac [("P",P)] expand_case i THEN
        Simp_tac (i+1) THEN
        Simp_tac i;

    (*This lemma restricts the effect of the rewrite rule u=v to the left-hand
      side of an equality.  Used in {Integ,Real}/simproc.ML*)
    Goal "x=y ==> (x=z) = (y=z)";
    by (asm_simp_tac HOL_ss 1);
    qed "restrict_to_left";

    (* default simpset *)
    val simpsetup =
      [fn thy => (simpset_ref_of thy := HOL_ss addcongs [if_weak_cong]; thy)];

    (*** integration of simplifier with classical reasoner ***)

    structure Clasimp = ClasimpFun
     (structure Simplifier = Simplifier and Splitter = Splitter
      and Classical  = Classical and Blast = Blast
      val iffD1 = iffD1 val iffD2 = iffD2 val notE = notE
      val cla_make_elim = cla_make_elim);
    open Clasimp;

    val HOL_css = (HOL_cs, HOL_ss);

    (*** A general refutation procedure ***)

    (* Parameters:

       test: term -> bool
       tests if a term is at all relevant to the refutation proof;
       if not, then it can be discarded. Can improve performance,
       esp. if disjunctions can be discarded (no case distinction needed!).

       prep_tac: int -> tactic
       A preparation tactic to be applied to the goal once all relevant premises
       have been moved to the conclusion.

       ref_tac: int -> tactic
       the actual refutation tactic. Should be able to deal with goals
       [| A1; ...; An |] ==> False
       where the Ai are atomic, i.e. no top-level &, | or EX
    *)

    local
      val nnf_simps =
            [imp_conv_disj,iff_conv_conj_imp,de_Morgan_disj,de_Morgan_conj,
             not_all,not_ex,not_not];
      val nnf_simpset =
            empty_ss setmkeqTrue mk_eq_True
                     setmksimps (mksimps mksimps_pairs)
                     addsimps nnf_simps;
      val prem_nnf_tac = full_simp_tac nnf_simpset
    in
    fun refute_tac test prep_tac ref_tac =
      let val refute_prems_tac =
            REPEAT_DETERM
                  (eresolve_tac [conjE, exE] 1 ORELSE
                   filter_prems_tac test 1 ORELSE
                   etac disjE 1) THEN
            ((etac notE 1 THEN eq_assume_tac 1) ORELSE
             ref_tac 1);
      in EVERY'[TRY o filter_prems_tac test,
                REPEAT_DETERM o etac rev_mp, prep_tac, rtac ccontr, prem_nnf_tac,
                SELECT_GOAL (DEPTH_SOLVE refute_prems_tac)]
      end;
    end;