src/HOL/simpdata.ML
author clasohm
Wed Mar 13 11:55:25 1996 +0100 (1996-03-13 ago)
changeset 1574 5a63ab90ee8a
parent 1548 afe750876848
child 1655 5be64540f275
permissions -rw-r--r--
modified primrec so it can be used in MiniML/Type.thy
     1 (*  Title:      HOL/simpdata.ML
     2     ID:         $Id$
     3     Author:     Tobias Nipkow
     4     Copyright   1991  University of Cambridge
     5 
     6 Instantiation of the generic simplifier
     7 *)
     8 
     9 open Simplifier;
    10 
    11 local
    12 
    13 fun prover s = prove_goal HOL.thy s (fn _ => [fast_tac HOL_cs 1]);
    14 
    15 val P_imp_P_iff_True = prover "P --> (P = True)" RS mp;
    16 val P_imp_P_eq_True = P_imp_P_iff_True RS eq_reflection;
    17 
    18 val not_P_imp_P_iff_F = prover "~P --> (P = False)" RS mp;
    19 val not_P_imp_P_eq_False = not_P_imp_P_iff_F RS eq_reflection;
    20 
    21 fun atomize pairs =
    22   let fun atoms th =
    23         (case concl_of th of
    24            Const("Trueprop",_) $ p =>
    25              (case head_of p of
    26                 Const(a,_) =>
    27                   (case assoc(pairs,a) of
    28                      Some(rls) => flat (map atoms ([th] RL rls))
    29                    | None => [th])
    30               | _ => [th])
    31          | _ => [th])
    32   in atoms end;
    33 
    34 fun mk_meta_eq r = case concl_of r of
    35         Const("==",_)$_$_ => r
    36     |   _$(Const("op =",_)$_$_) => r RS eq_reflection
    37     |   _$(Const("not",_)$_) => r RS not_P_imp_P_eq_False
    38     |   _ => r RS P_imp_P_eq_True;
    39 (* last 2 lines requires all formulae to be of the from Trueprop(.) *)
    40 
    41 fun gen_all th = forall_elim_vars (#maxidx(rep_thm th)+1) th;
    42 
    43 val imp_cong = impI RSN
    44     (2, prove_goal HOL.thy "(P=P')--> (P'--> (Q=Q'))--> ((P-->Q) = (P'-->Q'))"
    45         (fn _=> [fast_tac HOL_cs 1]) RS mp RS mp);
    46 
    47 val o_apply = prove_goalw HOL.thy [o_def] "(f o g)(x) = f(g(x))"
    48  (fn _ => [rtac refl 1]);
    49 
    50 val simp_thms = map prover
    51  [ "(x=x) = True",
    52    "(~True) = False", "(~False) = True", "(~ ~ P) = P",
    53    "(~P) ~= P", "P ~= (~P)", "(P ~= Q) = (P = (~Q))",
    54    "(True=P) = P", "(P=True) = P",
    55    "(True --> P) = P", "(False --> P) = True", 
    56    "(P --> True) = True", "(P --> P) = True",
    57    "(P --> False) = (~P)", "(P --> ~P) = (~P)",
    58    "(P & True) = P", "(True & P) = P", 
    59    "(P & False) = False", "(False & P) = False", "(P & P) = P",
    60    "(P | True) = True", "(True | P) = True", 
    61    "(P | False) = P", "(False | P) = P", "(P | P) = P",
    62    "(!x.P) = P", "(? x.P) = P", "? x. x=t", "(? x. x=t & P(x)) = P(t)",
    63    "(P|Q --> R) = ((P-->R)&(Q-->R))" ];
    64 
    65 in
    66 
    67 val meta_eq_to_obj_eq = prove_goal HOL.thy "x==y ==> x=y"
    68   (fn [prem] => [rewtac prem, rtac refl 1]);
    69 
    70 val eq_sym_conv = prover "(x=y) = (y=x)";
    71 
    72 val conj_assoc = prover "((P&Q)&R) = (P&(Q&R))";
    73 
    74 val if_True = prove_goalw HOL.thy [if_def] "(if True then x else y) = x"
    75  (fn _=>[fast_tac (HOL_cs addIs [select_equality]) 1]);
    76 
    77 val if_False = prove_goalw HOL.thy [if_def] "(if False then x else y) = y"
    78  (fn _=>[fast_tac (HOL_cs addIs [select_equality]) 1]);
    79 
    80 val if_P = prove_goal HOL.thy "P ==> (if P then x else y) = x"
    81  (fn [prem] => [ stac (prem RS eqTrueI) 1, rtac if_True 1 ]);
    82 
    83 val if_not_P = prove_goal HOL.thy "~P ==> (if P then x else y) = y"
    84  (fn [prem] => [ stac (prem RS not_P_imp_P_iff_F) 1, rtac if_False 1 ]);
    85 
    86 val expand_if = prove_goal HOL.thy
    87     "P(if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))"
    88  (fn _=> [ (res_inst_tac [("Q","Q")] (excluded_middle RS disjE) 1),
    89          rtac (if_P RS ssubst) 2,
    90          rtac (if_not_P RS ssubst) 1,
    91          REPEAT(fast_tac HOL_cs 1) ]);
    92 
    93 val if_bool_eq = prove_goal HOL.thy
    94                    "(if P then Q else R) = ((P-->Q) & (~P-->R))"
    95                    (fn _ => [rtac expand_if 1]);
    96 
    97 (*Add congruence rules for = (instead of ==) *)
    98 infix 4 addcongs;
    99 fun ss addcongs congs = ss addeqcongs (congs RL [eq_reflection]);
   100 
   101 fun Addcongs congs = (simpset := !simpset addcongs congs);
   102 
   103 (*Add a simpset to a classical set!*)
   104 infix 4 addss;
   105 fun cs addss ss = cs addbefore asm_full_simp_tac ss 1;
   106 
   107 val mksimps_pairs =
   108   [("op -->", [mp]), ("op &", [conjunct1,conjunct2]),
   109    ("All", [spec]), ("True", []), ("False", []),
   110    ("If", [if_bool_eq RS iffD1])];
   111 
   112 fun mksimps pairs = map mk_meta_eq o atomize pairs o gen_all;
   113 
   114 val HOL_ss = empty_ss
   115       setmksimps (mksimps mksimps_pairs)
   116       setsolver (fn prems => resolve_tac (TrueI::refl::prems) ORELSE' atac
   117                              ORELSE' etac FalseE)
   118       setsubgoaler asm_simp_tac
   119       addsimps ([if_True, if_False, o_apply, conj_assoc] @ simp_thms)
   120       addcongs [imp_cong];
   121 
   122 local val mktac = mk_case_split_tac (meta_eq_to_obj_eq RS iffD2)
   123 in
   124 fun split_tac splits = mktac (map mk_meta_eq splits)
   125 end;
   126 
   127 
   128 (* eliminiation of existential quantifiers in assumptions *)
   129 
   130 val ex_all_equiv =
   131   let val lemma1 = prove_goal HOL.thy
   132         "(? x. P(x) ==> PROP Q) ==> (!!x. P(x) ==> PROP Q)"
   133         (fn prems => [resolve_tac prems 1, etac exI 1]);
   134       val lemma2 = prove_goalw HOL.thy [Ex_def]
   135         "(!!x. P(x) ==> PROP Q) ==> (? x. P(x) ==> PROP Q)"
   136         (fn prems => [REPEAT(resolve_tac prems 1)])
   137   in equal_intr lemma1 lemma2 end;
   138 
   139 (* '&' congruence rule: not included by default!
   140    May slow rewrite proofs down by as much as 50% *)
   141 
   142 val conj_cong = impI RSN
   143     (2, prove_goal HOL.thy "(P=P')--> (P'--> (Q=Q'))--> ((P&Q) = (P'&Q'))"
   144         (fn _=> [fast_tac HOL_cs 1]) RS mp RS mp);
   145 
   146 val rev_conj_cong = impI RSN
   147     (2, prove_goal HOL.thy "(Q=Q')--> (Q'--> (P=P'))--> ((P&Q) = (P'&Q'))"
   148         (fn _=> [fast_tac HOL_cs 1]) RS mp RS mp);
   149 
   150 (** 'if' congruence rules: neither included by default! *)
   151 
   152 (*Simplifies x assuming c and y assuming ~c*)
   153 val if_cong = prove_goal HOL.thy
   154   "[| b=c; c ==> x=u; ~c ==> y=v |] ==>\
   155 \  (if b then x else y) = (if c then u else v)"
   156   (fn rew::prems =>
   157    [stac rew 1, stac expand_if 1, stac expand_if 1,
   158     fast_tac (HOL_cs addDs prems) 1]);
   159 
   160 (*Prevents simplification of x and y: much faster*)
   161 val if_weak_cong = prove_goal HOL.thy
   162   "b=c ==> (if b then x else y) = (if c then x else y)"
   163   (fn [prem] => [rtac (prem RS arg_cong) 1]);
   164 
   165 (*Prevents simplification of t: much faster*)
   166 val let_weak_cong = prove_goal HOL.thy
   167   "a = b ==> (let x=a in t(x)) = (let x=b in t(x))"
   168   (fn [prem] => [rtac (prem RS arg_cong) 1]);
   169 
   170 end;
   171 
   172 fun prove nm thm  = qed_goal nm HOL.thy thm (fn _ => [fast_tac HOL_cs 1]);
   173 
   174 prove "conj_commute" "(P&Q) = (Q&P)";
   175 prove "conj_left_commute" "(P&(Q&R)) = (Q&(P&R))";
   176 val conj_comms = [conj_commute, conj_left_commute];
   177 
   178 prove "conj_disj_distribL" "(P&(Q|R)) = (P&Q | P&R)";
   179 prove "conj_disj_distribR" "((P|Q)&R) = (P&R | Q&R)";
   180 
   181 prove "de_Morgan_disj" "(~(P | Q)) = (~P & ~Q)";
   182 prove "de_Morgan_conj" "(~(P & Q)) = (~P | ~Q)";
   183