src/FOL/simpdata.ML

Removal of (EX x. P) <-> P and (ALL x. P) <-> P
from ex_simps and all_simps. as they are already in quant_simps.

(*  Title:      FOL/simpdata
    ID:         $Id$
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1994  University of Cambridge

Simplification data for FOL
*)

(*** Rewrite rules ***)

fun int_prove_fun s = 
 (writeln s;  
  prove_goal IFOL.thy s
   (fn prems => [ (cut_facts_tac prems 1), 
                  (Int.fast_tac 1) ]));

val conj_simps = map int_prove_fun
 ["P & True <-> P",      "True & P <-> P",
  "P & False <-> False", "False & P <-> False",
  "P & P <-> P",
  "P & ~P <-> False",    "~P & P <-> False",
  "(P & Q) & R <-> P & (Q & R)"];

val disj_simps = map int_prove_fun
 ["P | True <-> True",  "True | P <-> True",
  "P | False <-> P",    "False | P <-> P",
  "P | P <-> P",
  "(P | Q) | R <-> P | (Q | R)"];

val not_simps = map int_prove_fun
 ["~(P|Q)  <-> ~P & ~Q",
  "~ False <-> True",   "~ True <-> False"];

val imp_simps = map int_prove_fun
 ["(P --> False) <-> ~P",       "(P --> True) <-> True",
  "(False --> P) <-> True",     "(True --> P) <-> P", 
  "(P --> P) <-> True",         "(P --> ~P) <-> ~P"];

val iff_simps = map int_prove_fun
 ["(True <-> P) <-> P",         "(P <-> True) <-> P",
  "(P <-> P) <-> True",
  "(False <-> P) <-> ~P",       "(P <-> False) <-> ~P"];

val quant_simps = map int_prove_fun
 ["(ALL x.P) <-> P",    "(EX x.P) <-> P"];

(*These are NOT supplied by default!*)
val distrib_simps  = map int_prove_fun
 ["P & (Q | R) <-> P&Q | P&R", 
  "(Q | R) & P <-> Q&P | R&P",
  "(P | Q --> R) <-> (P --> R) & (Q --> R)"];

(** Conversion into rewrite rules **)

fun gen_all th = forall_elim_vars (#maxidx(rep_thm th)+1) th;

(*Make atomic rewrite rules*)
fun atomize r =
  case concl_of r of
    Const("Trueprop",_) $ p =>
      (case p of
         Const("op -->",_)$_$_ => atomize(r RS mp)
       | Const("op &",_)$_$_   => atomize(r RS conjunct1) @
                                  atomize(r RS conjunct2)
       | Const("All",_)$_      => atomize(r RS spec)
       | Const("True",_)       => []    (*True is DELETED*)
       | Const("False",_)      => []    (*should False do something?*)
       | _                     => [r])
  | _ => [r];


val P_iff_F = int_prove_fun "~P ==> (P <-> False)";
val iff_reflection_F = P_iff_F RS iff_reflection;

val P_iff_T = int_prove_fun "P ==> (P <-> True)";
val iff_reflection_T = P_iff_T RS iff_reflection;

(*Make meta-equalities.  The operator below is Trueprop*)
fun mk_meta_eq th = case concl_of th of
    Const("==",_)$_$_           => th
  | _ $ (Const("op =",_)$_$_)   => th RS eq_reflection
  | _ $ (Const("op <->",_)$_$_) => th RS iff_reflection
  | _ $ (Const("Not",_)$_)      => th RS iff_reflection_F
  | _                           => th RS iff_reflection_T;

structure Induction = InductionFun(struct val spec=IFOL.spec end);

open Simplifier Induction;

(*Add congruence rules for = or <-> (instead of ==) *)
infix 4 addcongs;
fun ss addcongs congs =
    ss addeqcongs (congs RL [eq_reflection,iff_reflection]);

(*Add a simpset to a classical set!*)
infix 4 addss;
fun cs addss ss = cs addbefore asm_full_simp_tac ss 1;


val IFOL_simps =
   [refl RS P_iff_T] @ conj_simps @ disj_simps @ not_simps @ 
    imp_simps @ iff_simps @ quant_simps;

val notFalseI = int_prove_fun "~False";
val triv_rls = [TrueI,refl,iff_refl,notFalseI];

val IFOL_ss = 
  empty_ss 
  setmksimps (map mk_meta_eq o atomize o gen_all)
  setsolver  (fn prems => resolve_tac (triv_rls@prems) 
                          ORELSE' assume_tac
                          ORELSE' etac FalseE)
  setsubgoaler asm_simp_tac
  addsimps IFOL_simps
  addcongs [imp_cong];

(*Classical version...*)
fun prove_fun s = 
 (writeln s;  
  prove_goal FOL.thy s
   (fn prems => [ (cut_facts_tac prems 1), 
                  (Cla.fast_tac FOL_cs 1) ]));

(*Avoids duplication of subgoals after expand_if, when the true and false 
  cases boil down to the same thing.*) 
val cases_simp = prove_fun "(P --> Q) & (~P --> Q) <-> Q";

val cla_simps = 
    cases_simp::
    map prove_fun
     ["~(P&Q)  <-> ~P | ~Q",
      "P | ~P",             "~P | P",
      "~ ~ P <-> P",        "(~P --> P) <-> P",
      "(~P <-> ~Q) <-> (P<->Q)"];

(*At present, miniscoping is for classical logic only.  We do NOT include
  distribution of ALL over &, or dually that of EX over |.*)

(*Miniscoping: pushing in existential quantifiers*)
val ex_simps = map prove_fun 
                ["(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 prove_fun
                ["(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))"];

val FOL_ss = IFOL_ss addsimps (cla_simps @ ex_simps @ all_simps);

fun int_prove nm thm  = qed_goal nm IFOL.thy thm
    (fn prems => [ (cut_facts_tac prems 1), 
                   (Int.fast_tac 1) ]);

fun prove nm thm  = qed_goal nm FOL.thy thm (fn _ => [fast_tac FOL_cs 1]);

int_prove "conj_commute" "P&Q <-> Q&P";
int_prove "conj_left_commute" "P&(Q&R) <-> Q&(P&R)";
val conj_comms = [conj_commute, conj_left_commute];

int_prove "disj_commute" "P|Q <-> Q|P";
int_prove "disj_left_commute" "P|(Q|R) <-> Q|(P|R)";
val disj_comms = [disj_commute, disj_left_commute];

int_prove "conj_disj_distribL" "P&(Q|R) <-> (P&Q | P&R)";
int_prove "conj_disj_distribR" "(P|Q)&R <-> (P&R | Q&R)";

int_prove "disj_conj_distribL" "P|(Q&R) <-> (P|Q) & (P|R)";
int_prove "disj_conj_distribR" "(P&Q)|R <-> (P|R) & (Q|R)";

int_prove "imp_conj_distrib" "(P --> (Q&R)) <-> (P-->Q) & (P-->R)";
int_prove "imp_conj"         "((P&Q)-->R)   <-> (P --> (Q --> R))";
int_prove "imp_disj"         "(P|Q --> R)   <-> (P-->R) & (Q-->R)";

int_prove "de_Morgan_disj" "(~(P | Q)) <-> (~P & ~Q)";
prove     "de_Morgan_conj" "(~(P & Q)) <-> (~P | ~Q)";

prove     "not_iff" "~(P <-> Q) <-> (P <-> ~Q)";

prove     "not_all" "(~ (ALL x.P(x))) <-> (EX x.~P(x))";
prove     "imp_all" "((ALL x.P(x)) --> Q) <-> (EX x.P(x) --> Q)";
int_prove "not_ex"  "(~ (EX x.P(x))) <-> (ALL x.~P(x))";
int_prove "imp_ex" "((EX x. P(x)) --> Q) <-> (ALL x. P(x) --> Q)";

int_prove "ex_disj_distrib"
    "(EX x. P(x) | Q(x)) <-> ((EX x. P(x)) | (EX x. Q(x)))";
int_prove "all_conj_distrib"
    "(ALL x. P(x) & Q(x)) <-> ((ALL x. P(x)) & (ALL x. Q(x)))";


(*Used in ZF, perhaps elsewhere?*)
val meta_eq_to_obj_eq = prove_goal IFOL.thy "x==y ==> x=y"
  (fn [prem] => [rewtac prem, rtac refl 1]);

(*** case splitting ***)

qed_goal "meta_iffD" IFOL.thy "[| P==Q; Q |] ==> P"
        (fn [prem1,prem2] => [rewtac prem1, rtac prem2 1]);

local val mktac = mk_case_split_tac meta_iffD
in
fun split_tac splits = mktac (map mk_meta_eq splits)
end;

local val mktac = mk_case_split_inside_tac meta_iffD
in
fun split_inside_tac splits = mktac (map mk_meta_eq splits)
end;