src/FOL/simpdata.ML
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(*  Title:      FOL/simpdata.ML
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1994  University of Cambridge
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Simplification data for FOL.
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*)
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(* Elimination of True from asumptions: *)
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bind_thm ("True_implies_equals", prove_goal IFOL.thy
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 "(True ==> PROP P) == PROP P"
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(K [rtac equal_intr_rule 1, atac 2,
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          METAHYPS (fn prems => resolve_tac prems 1) 1,
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          rtac TrueI 1]));
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(*** Rewrite rules ***)
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fun int_prove_fun s =
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 (writeln s;
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  prove_goal IFOL.thy s
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   (fn prems => [ (cut_facts_tac prems 1),
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                  (IntPr.fast_tac 1) ]));
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bind_thms ("conj_simps", map int_prove_fun
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 ["P & True <-> P",      "True & P <-> P",
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  "P & False <-> False", "False & P <-> False",
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  "P & P <-> P", "P & P & Q <-> P & Q",
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  "P & ~P <-> False",    "~P & P <-> False",
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  "(P & Q) & R <-> P & (Q & R)"]);
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bind_thms ("disj_simps", map int_prove_fun
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 ["P | True <-> True",  "True | P <-> True",
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  "P | False <-> P",    "False | P <-> P",
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  "P | P <-> P", "P | P | Q <-> P | Q",
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  "(P | Q) | R <-> P | (Q | R)"]);
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bind_thms ("not_simps", map int_prove_fun
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 ["~(P|Q)  <-> ~P & ~Q",
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  "~ False <-> True",   "~ True <-> False"]);
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bind_thms ("imp_simps", map int_prove_fun
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 ["(P --> False) <-> ~P",       "(P --> True) <-> True",
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  "(False --> P) <-> True",     "(True --> P) <-> P",
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  "(P --> P) <-> True",         "(P --> ~P) <-> ~P"]);
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bind_thms ("iff_simps", map int_prove_fun
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 ["(True <-> P) <-> P",         "(P <-> True) <-> P",
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  "(P <-> P) <-> True",
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  "(False <-> P) <-> ~P",       "(P <-> False) <-> ~P"]);
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(*The x=t versions are needed for the simplification procedures*)
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bind_thms ("quant_simps", map int_prove_fun
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 ["(ALL x. P) <-> P",
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  "(ALL x. x=t --> P(x)) <-> P(t)",
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  "(ALL x. t=x --> P(x)) <-> P(t)",
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  "(EX x. P) <-> P",
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  "(EX x. x=t & P(x)) <-> P(t)",
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  "(EX x. t=x & P(x)) <-> P(t)"]);
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(*These are NOT supplied by default!*)
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bind_thms ("distrib_simps", map int_prove_fun
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 ["P & (Q | R) <-> P&Q | P&R",
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  "(Q | R) & P <-> Q&P | R&P",
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  "(P | Q --> R) <-> (P --> R) & (Q --> R)"]);
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(** Conversion into rewrite rules **)
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fun gen_all th = forall_elim_vars (#maxidx(rep_thm th)+1) th;
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bind_thm ("P_iff_F", int_prove_fun "~P ==> (P <-> False)");
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bind_thm ("iff_reflection_F", P_iff_F RS iff_reflection);
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bind_thm ("P_iff_T", int_prove_fun "P ==> (P <-> True)");
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bind_thm ("iff_reflection_T", P_iff_T RS iff_reflection);
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(*Make meta-equalities.  The operator below is Trueprop*)
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fun mk_meta_eq th = case concl_of th of
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    _ $ (Const("op =",_)$_$_)   => th RS eq_reflection
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  | _ $ (Const("op <->",_)$_$_) => th RS iff_reflection
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  | _                           =>
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  error("conclusion must be a =-equality or <->");;
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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("op <->",_)$_$_) => mk_meta_eq th
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  | _ $ (Const("Not",_)$_)      => th RS iff_reflection_F
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  | _                           => th RS iff_reflection_T;
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(*Replace premises x=y, X<->Y by X==Y*)
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val mk_meta_prems =
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    rule_by_tactic
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      (REPEAT_FIRST (resolve_tac [meta_eq_to_obj_eq, def_imp_iff]));
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(*Congruence rules for = or <-> (instead of ==)*)
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fun mk_meta_cong rl =
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  standard(mk_meta_eq (mk_meta_prems rl))
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  handle THM _ =>
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  error("Premises and conclusion of congruence rules must use =-equality or <->");
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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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(* ###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 = (map mk_eq o mk_atomize pairs o gen_all);
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(*** Classical laws ***)
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fun prove_fun s =
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 (writeln s;
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  prove_goal (the_context ()) s
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   (fn prems => [ (cut_facts_tac prems 1),
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                  (Cla.fast_tac FOL_cs 1) ]));
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(*Avoids duplication of subgoals after expand_if, when the true and false
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  cases boil down to the same thing.*)
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bind_thm ("cases_simp", prove_fun "(P --> Q) & (~P --> Q) <-> Q");
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(*** Miniscoping: pushing quantifiers in
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     We do NOT distribute of ALL over &, or dually that of EX over |
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     Baaz and Leitsch, On Skolemization and Proof Complexity (1994)
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     show that this step can increase proof length!
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***)
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(*existential miniscoping*)
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bind_thms ("int_ex_simps", map int_prove_fun
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 ["(EX x. P(x) & Q) <-> (EX x. P(x)) & Q",
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  "(EX x. P & Q(x)) <-> P & (EX x. Q(x))",
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  "(EX x. P(x) | Q) <-> (EX x. P(x)) | Q",
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  "(EX x. P | Q(x)) <-> P | (EX x. Q(x))"]);
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(*classical rules*)
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bind_thms ("cla_ex_simps", map prove_fun
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 ["(EX x. P(x) --> Q) <-> (ALL x. P(x)) --> Q",
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  "(EX x. P --> Q(x)) <-> P --> (EX x. Q(x))"]);
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bind_thms ("ex_simps", int_ex_simps @ cla_ex_simps);
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(*universal miniscoping*)
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bind_thms ("int_all_simps", map int_prove_fun
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 ["(ALL x. P(x) & Q) <-> (ALL x. P(x)) & Q",
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  "(ALL x. P & Q(x)) <-> P & (ALL x. Q(x))",
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  "(ALL x. P(x) --> Q) <-> (EX x. P(x)) --> Q",
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  "(ALL x. P --> Q(x)) <-> P --> (ALL x. Q(x))"]);
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(*classical rules*)
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bind_thms ("cla_all_simps", map prove_fun
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 ["(ALL x. P(x) | Q) <-> (ALL x. P(x)) | Q",
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  "(ALL x. P | Q(x)) <-> P | (ALL x. Q(x))"]);
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bind_thms ("all_simps", int_all_simps @ cla_all_simps);
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(*** Named rewrite rules proved for IFOL ***)
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fun int_prove nm thm  = qed_goal nm IFOL.thy thm
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    (fn prems => [ (cut_facts_tac prems 1),
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                   (IntPr.fast_tac 1) ]);
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fun prove nm thm  = qed_goal nm (the_context ()) thm (fn _ => [Blast_tac 1]);
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int_prove "conj_commute" "P&Q <-> Q&P";
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int_prove "conj_left_commute" "P&(Q&R) <-> Q&(P&R)";
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bind_thms ("conj_comms", [conj_commute, conj_left_commute]);
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int_prove "disj_commute" "P|Q <-> Q|P";
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int_prove "disj_left_commute" "P|(Q|R) <-> Q|(P|R)";
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bind_thms ("disj_comms", [disj_commute, disj_left_commute]);
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int_prove "conj_disj_distribL" "P&(Q|R) <-> (P&Q | P&R)";
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int_prove "conj_disj_distribR" "(P|Q)&R <-> (P&R | Q&R)";
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int_prove "disj_conj_distribL" "P|(Q&R) <-> (P|Q) & (P|R)";
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int_prove "disj_conj_distribR" "(P&Q)|R <-> (P|R) & (Q|R)";
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int_prove "imp_conj_distrib" "(P --> (Q&R)) <-> (P-->Q) & (P-->R)";
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int_prove "imp_conj"         "((P&Q)-->R)   <-> (P --> (Q --> R))";
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int_prove "imp_disj"         "(P|Q --> R)   <-> (P-->R) & (Q-->R)";
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prove "imp_disj1" "(P-->Q) | R <-> (P-->Q | R)";
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prove "imp_disj2" "Q | (P-->R) <-> (P-->Q | R)";
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int_prove "de_Morgan_disj" "(~(P | Q)) <-> (~P & ~Q)";
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prove     "de_Morgan_conj" "(~(P & Q)) <-> (~P | ~Q)";
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prove     "not_iff" "~(P <-> Q) <-> (P <-> ~Q)";
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prove     "not_all" "(~ (ALL x. P(x))) <-> (EX x.~P(x))";
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prove     "imp_all" "((ALL x. P(x)) --> Q) <-> (EX x. P(x) --> Q)";
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int_prove "not_ex"  "(~ (EX x. P(x))) <-> (ALL x.~P(x))";
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int_prove "imp_ex" "((EX x. P(x)) --> Q) <-> (ALL x. P(x) --> Q)";
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int_prove "ex_disj_distrib"
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    "(EX x. P(x) | Q(x)) <-> ((EX x. P(x)) | (EX x. Q(x)))";
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int_prove "all_conj_distrib"
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    "(ALL x. P(x) & Q(x)) <-> ((ALL x. P(x)) & (ALL x. Q(x)))";
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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()) "ALL x. P(x) <-> Q(x) ==> (ALL x. P(x)) <-> (ALL 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 = FOLogic.conj
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  val imp  = FOLogic.imp
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  (*rules*)
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  val iff_reflection = iff_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 =
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  read_cterm (Theory.sign_of (the_context ())) ("EX x. P(x) & Q(x)", FOLogic.oT)
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val all_pattern =
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  read_cterm (Theory.sign_of (the_context ())) ("ALL x. P(x) --> Q(x)", FOLogic.oT)
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in
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val defEX_regroup =
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  mk_simproc "defined EX" [ex_pattern] Quantifier1.rearrange_ex;
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val defALL_regroup =
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  mk_simproc "defined ALL" [all_pattern] Quantifier1.rearrange_all;
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end;
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(*** Case splitting ***)
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bind_thm ("meta_eq_to_iff", prove_goal IFOL.thy "x==y ==> x<->y"
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  (fn [prem] => [rewtac prem, rtac iffI 1, atac 1, atac 1]));
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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_iff
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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
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  val contrapos2     = contrapos2
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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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(*** Standard simpsets ***)
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structure Induction = InductionFun(struct val spec=IFOL.spec end);
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open Induction;
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bind_thms ("meta_simps",
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 [triv_forall_equality,   (* prunes params *)
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  True_implies_equals]);  (* prune asms `True' *)
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bind_thms ("IFOL_simps",
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 [refl RS P_iff_T] @ conj_simps @ disj_simps @ not_simps @
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  imp_simps @ iff_simps @ quant_simps);
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bind_thm ("notFalseI", int_prove_fun "~False");
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bind_thms ("triv_rls", [TrueI,refl,reflexive_thm,iff_refl,notFalseI]);
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fun unsafe_solver prems = FIRST'[resolve_tac (triv_rls@prems),
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                                 atac, etac FalseE];
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(*No premature instantiation of variables during simplification*)
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fun   safe_solver prems = FIRST'[match_tac (triv_rls@prems),
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                                 eq_assume_tac, ematch_tac [FalseE]];
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(*No simprules, but basic infastructure for simplification*)
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val FOL_basic_ss = empty_ss
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  setsubgoaler asm_simp_tac
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  setSSolver (mk_solver "FOL safe" safe_solver)
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  setSolver (mk_solver "FOL unsafe" unsafe_solver)
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  setmksimps (mksimps mksimps_pairs)
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  setmkcong mk_meta_cong;
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(*intuitionistic simprules only*)
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val IFOL_ss = FOL_basic_ss
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  addsimps (meta_simps @ IFOL_simps @ int_ex_simps @ int_all_simps)
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  addsimprocs [defALL_regroup, defEX_regroup]    
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  addcongs [imp_cong];
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bind_thms ("cla_simps",
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  [de_Morgan_conj, de_Morgan_disj, imp_disj1, imp_disj2,
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   not_all, not_ex, cases_simp] @
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  map prove_fun
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   ["~(P&Q) <-> ~P | ~Q",
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    "P | ~P",             "~P | P",
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    "~ ~ P <-> P",        "(~P --> P) <-> P",
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    "(~P <-> ~Q) <-> (P<->Q)"]);
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(*classical simprules too*)
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val FOL_ss = IFOL_ss addsimps (cla_simps @ cla_ex_simps @ cla_all_simps);
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val simpsetup = [fn thy => (simpset_ref_of thy := FOL_ss; 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  = Cla 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 FOL_css = (FOL_cs, FOL_ss);