(*  Title:      HOL/simpdata.ML

     ID:         $Id$

     Author:     Tobias Nipkow

     Copyright   1991  University of Cambridge

 Instantiation of the generic simplifier for HOL.

 *)

 section "Simplifier";

 val [prem] = goal (the_context ()) "x==y ==> x=y";

 by (rewtac prem);

 by (rtac refl 1);

 qed "meta_eq_to_obj_eq";

 Goal "(%s. f s) = f";

 br refl 1;

 qed "eta_contract_eq";

 local

   fun prover s = prove_goal (the_context ()) s (fn _ => [(Blast_tac 1)]);

 in

 (*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;

 val Eq_TrueI  = mk_meta_eq(prover  "P --> (P = True)"  RS mp);

 val Eq_FalseI = mk_meta_eq(prover "~P --> (P = False)" RS mp);

 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;

 (* last 2 lines requires all formulae to be of the from Trueprop(.) *)

 fun mk_eq_True r = Some(r RS meta_eq_to_obj_eq RS Eq_TrueI);

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

 fun mk_meta_cong rl =

   standard(mk_meta_eq(replicate (nprems_of rl) meta_eq_to_obj_eq MRS rl))

   handle THM _ =>

   error("Premises and conclusion of congruence rules must be =-equalities");

 val not_not = prover "(~ ~ P) = P";

 val simp_thms = [not_not] @ map prover

  [ "(x=x) = True",

    "(~True) = False", "(~False) = True",

    "(~P) ~= P", "P ~= (~P)", "(P ~= Q) = (P = (~Q))",

    "(True=P) = P", "(P=True) = P", "(False=P) = (~P)", "(P=False) = (~P)",

    "(True --> P) = P", "(False --> P) = True",

    "(P --> True) = True", "(P --> P) = True",

    "(P --> False) = (~P)", "(P --> ~P) = (~P)",

    "(P & True) = P", "(True & P) = P",

    "(P & False) = False", "(False & P) = False",

    "(P & P) = P", "(P & (P & Q)) = (P & Q)",

    "(P & ~P) = False",    "(~P & P) = False",

    "(P | True) = True", "(True | P) = True",

    "(P | False) = P", "(False | P) = P",

    "(P | P) = P", "(P | (P | Q)) = (P | Q)",

    "(P | ~P) = True",    "(~P | P) = True",

    "((~P) = (~Q)) = (P=Q)",

    "(!x. P) = P", "(? x. P) = P", "? x. x=t", "? x. t=x",

 (*two needed for the one-point-rule quantifier simplification procs*)

    "(? x. x=t & P(x)) = P(t)",          (*essential for termination!!*)

    "(! x. t=x --> P(x)) = P(t)" ];      (*covers a stray case*)

 val imp_cong = standard(impI RSN

     (2, prove_goal (the_context ()) "(P=P')--> (P'--> (Q=Q'))--> ((P-->Q) = (P'-->Q'))"

         (fn _=> [(Blast_tac 1)]) RS mp RS mp));

 (*Miniscoping: pushing in existential quantifiers*)

 val ex_simps = map prover

                 ["(EX x. P x & Q)   = ((EX x. P x) & Q)",

                  "(EX x. P & Q x)   = (P & (EX x. Q x))",

                  "(EX x. P x | Q)   = ((EX x. P x) | Q)",

                  "(EX x. P | Q x)   = (P | (EX x. Q x))",

                  "(EX x. P x --> Q) = ((ALL x. P x) --> Q)",

                  "(EX x. P --> Q x) = (P --> (EX x. Q x))"];

 (*Miniscoping: pushing in universal quantifiers*)

 val all_simps = map prover

                 ["(ALL x. P x & Q)   = ((ALL x. P x) & Q)",

                  "(ALL x. P & Q x)   = (P & (ALL x. Q x))",

                  "(ALL x. P x | Q)   = ((ALL x. P x) | Q)",

                  "(ALL x. P | Q x)   = (P | (ALL x. Q x))",

                  "(ALL x. P x --> Q) = ((EX x. P x) --> Q)",

                  "(ALL x. P --> Q x) = (P --> (ALL x. Q x))"];

 (* elimination of existential quantifiers in assumptions *)

 val ex_all_equiv =

   let val lemma1 = prove_goal (the_context ())

         "(? x. P(x) ==> PROP Q) ==> (!!x. P(x) ==> PROP Q)"

         (fn prems => [resolve_tac prems 1, etac exI 1]);

       val lemma2 = prove_goalw (the_context ()) [Ex_def]

         "(!!x. P(x) ==> PROP Q) ==> (? x. P(x) ==> PROP Q)"

         (fn prems => [(REPEAT(resolve_tac prems 1))])

   in equal_intr lemma1 lemma2 end;

 end;

 bind_thms ("ex_simps", ex_simps);

 bind_thms ("all_simps", all_simps);

 bind_thm ("not_not", not_not);

 bind_thm ("imp_cong", imp_cong);

 (* Elimination of True from asumptions: *)

 val True_implies_equals = prove_goal (the_context ())

  "(True ==> PROP P) == PROP P"

 (fn _ => [rtac equal_intr_rule 1, atac 2,

           METAHYPS (fn prems => resolve_tac prems 1) 1,

           rtac TrueI 1]);

 fun prove nm thm  = qed_goal nm (the_context ()) thm (fn _ => [(Blast_tac 1)]);

 prove "eq_commute" "(a=b) = (b=a)";

 prove "eq_left_commute" "(P=(Q=R)) = (Q=(P=R))";

 prove "eq_assoc" "((P=Q)=R) = (P=(Q=R))";

 val eq_ac = [eq_commute, eq_left_commute, eq_assoc];

 prove "neq_commute" "(a~=b) = (b~=a)";

 prove "conj_commute" "(P&Q) = (Q&P)";

 prove "conj_left_commute" "(P&(Q&R)) = (Q&(P&R))";

 val conj_comms = [conj_commute, conj_left_commute];

 prove "conj_assoc" "((P&Q)&R) = (P&(Q&R))";

 prove "disj_commute" "(P|Q) = (Q|P)";

 prove "disj_left_commute" "(P|(Q|R)) = (Q|(P|R))";

 val disj_comms = [disj_commute, disj_left_commute];

 prove "disj_assoc" "((P|Q)|R) = (P|(Q|R))";

 prove "conj_disj_distribL" "(P&(Q|R)) = (P&Q | P&R)";

 prove "conj_disj_distribR" "((P|Q)&R) = (P&R | Q&R)";

 prove "disj_conj_distribL" "(P|(Q&R)) = ((P|Q) & (P|R))";

 prove "disj_conj_distribR" "((P&Q)|R) = ((P|R) & (Q|R))";

 prove "imp_conjR" "(P --> (Q&R)) = ((P-->Q) & (P-->R))";

 prove "imp_conjL" "((P&Q) -->R)  = (P --> (Q --> R))";

 prove "imp_disjL" "((P|Q) --> R) = ((P-->R)&(Q-->R))";

 (*These two are specialized, but imp_disj_not1 is useful in Auth/Yahalom.ML*)

 prove "imp_disj_not1" "(P --> Q | R) = (~Q --> P --> R)";

 prove "imp_disj_not2" "(P --> Q | R) = (~R --> P --> Q)";

 prove "imp_disj1" "((P-->Q)|R) = (P--> Q|R)";

 prove "imp_disj2" "(Q|(P-->R)) = (P--> Q|R)";

 prove "de_Morgan_disj" "(~(P | Q)) = (~P & ~Q)";

 prove "de_Morgan_conj" "(~(P & Q)) = (~P | ~Q)";

 prove "not_imp" "(~(P --> Q)) = (P & ~Q)";

 prove "not_iff" "(P~=Q) = (P = (~Q))";

 prove "disj_not1" "(~P | Q) = (P --> Q)";

 prove "disj_not2" "(P | ~Q) = (Q --> P)"; (* changes orientation :-( *)

 prove "imp_conv_disj" "(P --> Q) = ((~P) | Q)";

 prove "iff_conv_conj_imp" "(P = Q) = ((P --> Q) & (Q --> P))";

 (*Avoids duplication of subgoals after split_if, when the true and false

   cases boil down to the same thing.*)

 prove "cases_simp" "((P --> Q) & (~P --> Q)) = Q";

 prove "not_all" "(~ (! x. P(x))) = (? x.~P(x))";

 prove "imp_all" "((! x. P x) --> Q) = (? x. P x --> Q)";

 prove "not_ex"  "(~ (? x. P(x))) = (! x.~P(x))";

 prove "imp_ex" "((? x. P x) --> Q) = (! x. P x --> Q)";

 prove "ex_disj_distrib" "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))";

 prove "all_conj_distrib" "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))";

 (* '&' congruence rule: not included by default!

    May slow rewrite proofs down by as much as 50% *)

 let val th = prove_goal (the_context ())

                 "(P=P')--> (P'--> (Q=Q'))--> ((P&Q) = (P'&Q'))"

                 (fn _=> [(Blast_tac 1)])

 in  bind_thm("conj_cong",standard (impI RSN (2, th RS mp RS mp)))  end;

 let val th = prove_goal (the_context ())

                 "(Q=Q')--> (Q'--> (P=P'))--> ((P&Q) = (P'&Q'))"

                 (fn _=> [(Blast_tac 1)])

 in  bind_thm("rev_conj_cong",standard (impI RSN (2, th RS mp RS mp)))  end;

 (* '|' congruence rule: not included by default! *)

 let val th = prove_goal (the_context ())

                 "(P=P')--> (~P'--> (Q=Q'))--> ((P|Q) = (P'|Q'))"

                 (fn _=> [(Blast_tac 1)])

 in  bind_thm("disj_cong",standard (impI RSN (2, th RS mp RS mp)))  end;

 prove "eq_sym_conv" "(x=y) = (y=x)";

 (** if-then-else rules **)

 Goalw [if_def] "(if True then x else y) = x";

 by (Blast_tac 1);

 qed "if_True";

 Goalw [if_def] "(if False then x else y) = y";

 by (Blast_tac 1);

 qed "if_False";

 Goalw [if_def] "P ==> (if P then x else y) = x";

 by (Blast_tac 1);

 qed "if_P";

 Goalw [if_def] "~P ==> (if P then x else y) = y";

 by (Blast_tac 1);

 qed "if_not_P";

 Goal "P(if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))";

 by (res_inst_tac [("Q","Q")] (excluded_middle RS disjE) 1);

 by (stac if_P 2);

 by (stac if_not_P 1);

 by (ALLGOALS (Blast_tac));

 qed "split_if";

 Goal "P(if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))";

 by (stac split_if 1);

 by (Blast_tac 1);

 qed "split_if_asm";

 bind_thms ("if_splits", [split_if, split_if_asm]);

 bind_thm ("if_def2", read_instantiate [("P","\\<lambda>x. x")] split_if);

 Goal "(if c then x else x) = x";

 by (stac split_if 1);

 by (Blast_tac 1);

 qed "if_cancel";

 Goal "(if x = y then y else x) = x";

 by (stac split_if 1);

 by (Blast_tac 1);

 qed "if_eq_cancel";

 (*This form is useful for expanding IFs on the RIGHT of the ==> symbol*)

 Goal "(if P then Q else R) = ((P-->Q) & (~P-->R))";

 by (rtac split_if 1);

 qed "if_bool_eq_conj";

 (*And this form is useful for expanding IFs on the LEFT*)

 Goal "(if P then Q else R) = ((P&Q) | (~P&R))";

 by (stac split_if 1);

 by (Blast_tac 1);

 qed "if_bool_eq_disj";

 (*** 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;

   val conj = HOLogic.conj

   val imp  = HOLogic.imp

   (*rules*)

   val iff_reflection = eq_reflection

   val iffI = iffI

   val sym  = sym

   val conjI= conjI

   val conjE= conjE

   val impI = impI

   val impE = impE

   val mp   = mp

   val exI  = exI

   val exE  = exE

   val allI = allI

   val allE = allE

 end);

 local

 val ex_pattern =

   Thm.read_cterm (Theory.sign_of (the_context ())) ("EX x. P(x) & Q(x)",HOLogic.boolT)

 val all_pattern =

   Thm.read_cterm (Theory.sign_of (the_context ())) ("ALL x. P(x) & P'(x) --> Q(x)",HOLogic.boolT)

 in

 val defEX_regroup =

   mk_simproc "defined EX" [ex_pattern] Quantifier1.rearrange_ex;

 val defALL_regroup =

   mk_simproc "defined ALL" [all_pattern] Quantifier1.rearrange_all;

 end;

 (*** Case splitting ***)

 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;

 (*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 mksimps_pairs =

   [("op -->", [mp]), ("op &", [conjunct1,conjunct2]),

    ("All", [spec]), ("True", []), ("False", []),

    ("If", [if_bool_eq_conj RS iffD1])];

 (* ###FIXME: move to Provers/simplifier.ML

 val mk_atomize:      (string * thm list) list -> thm -> thm list

 *)

 (* ###FIXME: move to Provers/simplifier.ML *)

 fun mk_atomize pairs =

   let fun atoms th =

         (case concl_of th of

            Const("Trueprop",_) $ p =>

              (case head_of p of

                 Const(a,_) =>

                   (case assoc(pairs,a) of

                      Some(rls) => flat (map atoms ([th] RL rls))

                    | None => [th])

               | _ => [th])

          | _ => [th])

   in atoms end;

 fun mksimps pairs = (map mk_eq o mk_atomize pairs o forall_elim_vars_safe);

 fun unsafe_solver_tac prems =

   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 =

   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;

 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, some_eq_trivial, some_sym_eq_trivial,

        thm"plus_ac0.zero", thm"plus_ac0_zero_right"]

      @ ex_simps @ all_simps @ simp_thms)

      addsimprocs [defALL_regroup,defEX_regroup]

      addcongs [imp_cong]

      addsplits [split_if];

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

 fun hol_rewrite_cterm rews =

   #2 o Thm.dest_comb o #prop o Thm.crep_thm o Simplifier.full_rewrite (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";

 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 dest_Trueprop = HOLogic.dest_Trueprop

   val iff_const = HOLogic.eq_const HOLogic.boolT

   val not_const = HOLogic.not_const

   val notE = notE val iffD1 = iffD1 val iffD2 = iffD2

   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

 *)

 fun refute_tac test prep_tac ref_tac =

   let 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;

       val refute_prems_tac =

         REPEAT(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,

             DETERM o REPEAT o etac rev_mp, prep_tac, rtac ccontr, prem_nnf_tac,

             SELECT_GOAL (DEPTH_SOLVE refute_prems_tac)]

   end;