src/LCF/LCF.ML
author oheimb
Fri Jun 02 20:38:28 2000 +0200 (2000-06-02)
changeset 9028 8a1ec8f05f14
parent 3837 d7f033c74b38
permissions -rw-r--r--
added HOL/Prolog
     1 (*  Title:      LCF/lcf.ML
     2     ID:         $Id$
     3     Author:     Tobias Nipkow
     4     Copyright   1992  University of Cambridge
     5 
     6 For lcf.thy.  Basic lemmas about LCF
     7 *)
     8 
     9 open LCF;
    10 
    11 signature LCF_LEMMAS =
    12 sig
    13   val ap_term: thm
    14   val ap_thm: thm
    15   val COND_cases: thm
    16   val COND_cases_iff: thm
    17   val Contrapos: thm
    18   val cong: thm
    19   val ext: thm
    20   val eq_imp_less1: thm
    21   val eq_imp_less2: thm
    22   val less_anti_sym: thm
    23   val less_ap_term: thm
    24   val less_ap_thm: thm
    25   val less_refl: thm
    26   val less_UU: thm
    27   val not_UU_eq_TT: thm
    28   val not_UU_eq_FF: thm
    29   val not_TT_eq_UU: thm
    30   val not_TT_eq_FF: thm
    31   val not_FF_eq_UU: thm
    32   val not_FF_eq_TT: thm
    33   val rstac: thm list -> int -> tactic
    34   val stac: thm -> int -> tactic
    35   val sstac: thm list -> int -> tactic
    36   val strip_tac: int -> tactic
    37   val tr_induct: thm
    38   val UU_abs: thm
    39   val UU_app: thm
    40 end;
    41 
    42 
    43 structure LCF_Lemmas : LCF_LEMMAS =
    44 
    45 struct
    46 
    47 (* Standard abbreviations *)
    48 
    49 val rstac = resolve_tac;
    50 fun stac th = rtac(th RS sym RS subst);
    51 fun sstac ths = EVERY' (map stac ths);
    52 
    53 fun strip_tac i = REPEAT(rstac [impI,allI] i); 
    54 
    55 val eq_imp_less1 = prove_goal thy "x=y ==> x << y"
    56         (fn prems => [rtac (rewrite_rule[eq_def](hd prems) RS conjunct1) 1]);
    57 
    58 val eq_imp_less2 = prove_goal thy "x=y ==> y << x"
    59         (fn prems => [rtac (rewrite_rule[eq_def](hd prems) RS conjunct2) 1]);
    60 
    61 val less_refl = refl RS eq_imp_less1;
    62 
    63 val less_anti_sym = prove_goal thy "[| x << y; y << x |] ==> x=y"
    64         (fn prems => [rewtac eq_def,
    65                       REPEAT(rstac(conjI::prems)1)]);
    66 
    67 val ext = prove_goal thy
    68         "(!!x::'a::cpo. f(x)=(g(x)::'b::cpo)) ==> (%x. f(x))=(%x. g(x))"
    69         (fn [prem] => [REPEAT(rstac[less_anti_sym, less_ext, allI,
    70                                     prem RS eq_imp_less1,
    71                                     prem RS eq_imp_less2]1)]);
    72 
    73 val cong = prove_goal thy "[| f=g; x=y |] ==> f(x)=g(y)"
    74         (fn prems => [cut_facts_tac prems 1, etac subst 1, etac subst 1,
    75                       rtac refl 1]);
    76 
    77 val less_ap_term = less_refl RS mono;
    78 val less_ap_thm = less_refl RSN (2,mono);
    79 val ap_term = refl RS cong;
    80 val ap_thm = refl RSN (2,cong);
    81 
    82 val UU_abs = prove_goal thy "(%x::'a::cpo. UU) = UU"
    83         (fn _ => [rtac less_anti_sym 1, rtac minimal 2,
    84                   rtac less_ext 1, rtac allI 1, rtac minimal 1]);
    85 
    86 val UU_app = UU_abs RS sym RS ap_thm;
    87 
    88 val less_UU = prove_goal thy "x << UU ==> x=UU"
    89         (fn prems=> [rtac less_anti_sym 1,rstac prems 1,rtac minimal 1]);
    90 
    91 
    92 val tr_induct = prove_goal thy "[| P(UU); P(TT); P(FF) |] ==> ALL b. P(b)"
    93         (fn prems => [rtac allI 1, rtac mp 1,
    94                       res_inst_tac[("p","b")]tr_cases 2,
    95                       fast_tac (FOL_cs addIs prems) 1]);
    96 
    97 
    98 val Contrapos = prove_goal thy "(A ==> B) ==> (~B ==> ~A)"
    99         (fn prems => [rtac notI 1, rtac notE 1, rstac prems 1,
   100                       rstac prems 1, atac 1]);
   101 
   102 val not_less_imp_not_eq1 = eq_imp_less1 COMP Contrapos;
   103 val not_less_imp_not_eq2 = eq_imp_less2 COMP Contrapos;
   104 
   105 val not_UU_eq_TT = not_TT_less_UU RS not_less_imp_not_eq2;
   106 val not_UU_eq_FF = not_FF_less_UU RS not_less_imp_not_eq2;
   107 val not_TT_eq_UU = not_TT_less_UU RS not_less_imp_not_eq1;
   108 val not_TT_eq_FF = not_TT_less_FF RS not_less_imp_not_eq1;
   109 val not_FF_eq_UU = not_FF_less_UU RS not_less_imp_not_eq1;
   110 val not_FF_eq_TT = not_FF_less_TT RS not_less_imp_not_eq1;
   111 
   112 
   113 val COND_cases_iff = (prove_goal thy
   114   "ALL b. P(b=>x|y) <-> (b=UU-->P(UU)) & (b=TT-->P(x)) & (b=FF-->P(y))"
   115         (fn _ => [cut_facts_tac [not_UU_eq_TT,not_UU_eq_FF,not_TT_eq_UU,
   116                                  not_TT_eq_FF,not_FF_eq_UU,not_FF_eq_TT]1,
   117                   rtac tr_induct 1, stac COND_UU 1, stac COND_TT 2,
   118                   stac COND_FF 3, REPEAT(fast_tac FOL_cs 1)]))  RS spec;
   119 
   120 val lemma = prove_goal thy "A<->B ==> B ==> A"
   121         (fn prems => [cut_facts_tac prems 1, rewtac iff_def,
   122                       fast_tac FOL_cs 1]);
   123 
   124 val COND_cases = conjI RSN (2,conjI RS (COND_cases_iff RS lemma));
   125 
   126 end;
   127 
   128 open LCF_Lemmas;
   129