src/HOL/Hilbert_Choice.thy
author paulson
Wed Jul 25 13:13:01 2001 +0200 (2001-07-25)
changeset 11451 8abfb4f7bd02
child 11454 7514e5e21cb8
permissions -rw-r--r--
partial restructuring to reduce dependence on Axiom of Choice
     1 (*  Title:      HOL/Hilbert_Choice.thy
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson
     4     Copyright   2001  University of Cambridge
     5 
     6 Hilbert's epsilon-operator and everything to do with the Axiom of Choice
     7 *)
     8 
     9 theory Hilbert_Choice = NatArith
    10 files ("Hilbert_Choice_lemmas.ML") ("meson_lemmas.ML") ("Tools/meson.ML"):
    11 
    12 consts
    13   Eps           :: "('a => bool) => 'a"
    14 
    15 
    16 syntax (input)
    17   "_Eps"        :: "[pttrn, bool] => 'a"                 ("(3\\<epsilon>_./ _)" [0, 10] 10)
    18 
    19 syntax (HOL)
    20   "_Eps"        :: "[pttrn, bool] => 'a"                 ("(3@ _./ _)" [0, 10] 10)
    21 
    22 syntax
    23   "_Eps"        :: "[pttrn, bool] => 'a"                 ("(3SOME _./ _)" [0, 10] 10)
    24 
    25 translations
    26   "SOME x. P"             == "Eps (%x. P)"
    27 
    28 axioms  
    29   someI:        "P (x::'a) ==> P (SOME x. P x)"
    30 
    31 
    32 constdefs  
    33   inv :: "('a => 'b) => ('b => 'a)"
    34     "inv(f::'a=>'b) == % y. @x. f(x)=y"
    35 
    36   Inv :: "['a set, 'a => 'b] => ('b => 'a)"
    37     "Inv A f == (% x. (@ y. y : A & f y = x))"
    38 
    39 
    40 use "Hilbert_Choice_lemmas.ML"
    41 
    42 
    43 (** Least value operator **)
    44 
    45 constdefs
    46   LeastM   :: "['a => 'b::ord, 'a => bool] => 'a"
    47               "LeastM m P == @x. P x & (ALL y. P y --> m x <= m y)"
    48 
    49 syntax
    50  "@LeastM" :: "[pttrn, 'a=>'b::ord, bool] => 'a" ("LEAST _ WRT _. _" [0,4,10]10)
    51 
    52 translations
    53                 "LEAST x WRT m. P" == "LeastM m (%x. P)"
    54 
    55 lemma LeastMI2:
    56   "[| P x; !!y. P y ==> m x <= m y;
    57            !!x. [| P x; \\<forall>y. P y --> m x \\<le> m y |] ==> Q x |]
    58    ==> Q (LeastM m P)";
    59 apply (unfold LeastM_def)
    60 apply (rule someI2_ex)
    61 apply  blast
    62 apply blast
    63 done
    64 
    65 lemma LeastM_equality:
    66  "[| P k; !!x. P x ==> m k <= m x |] ==> m (LEAST x WRT m. P x) = 
    67      (m k::'a::order)";
    68 apply (rule LeastMI2)
    69 apply   assumption
    70 apply  blast
    71 apply (blast intro!: order_antisym) 
    72 done
    73 
    74 
    75 (** Greatest value operator **)
    76 
    77 constdefs
    78   GreatestM   :: "['a => 'b::ord, 'a => bool] => 'a"
    79               "GreatestM m P == @x. P x & (ALL y. P y --> m y <= m x)"
    80   
    81   Greatest    :: "('a::ord => bool) => 'a"         (binder "GREATEST " 10)
    82               "Greatest     == GreatestM (%x. x)"
    83 
    84 syntax
    85  "@GreatestM" :: "[pttrn, 'a=>'b::ord, bool] => 'a"
    86                                         ("GREATEST _ WRT _. _" [0,4,10]10)
    87 
    88 translations
    89               "GREATEST x WRT m. P" == "GreatestM m (%x. P)"
    90 
    91 lemma GreatestMI2:
    92      "[| P x;
    93 	 !!y. P y ==> m y <= m x;
    94          !!x. [| P x; \\<forall>y. P y --> m y \\<le> m x |] ==> Q x |]
    95       ==> Q (GreatestM m P)";
    96 apply (unfold GreatestM_def)
    97 apply (rule someI2_ex)
    98 apply  blast
    99 apply blast
   100 done
   101 
   102 lemma GreatestM_equality:
   103  "[| P k;  !!x. P x ==> m x <= m k |]
   104   ==> m (GREATEST x WRT m. P x) = (m k::'a::order)";
   105 apply (rule_tac m=m in GreatestMI2)
   106 apply   assumption
   107 apply  blast
   108 apply (blast intro!: order_antisym) 
   109 done
   110 
   111 lemma Greatest_equality:
   112   "[| P (k::'a::order); !!x. P x ==> x <= k |] ==> (GREATEST x. P x) = k";
   113 apply (unfold Greatest_def)
   114 apply (erule GreatestM_equality)
   115 apply blast
   116 done
   117 
   118 lemma ex_has_greatest_nat_lemma:
   119      "[|P k;  ALL x. P x --> (EX y. P y & ~ ((m y::nat) <= m x))|]  
   120       ==> EX y. P y & ~ (m y < m k + n)"
   121 apply (induct_tac "n")
   122 apply force
   123 (*ind step*)
   124 apply (force simp add: le_Suc_eq)
   125 done
   126 
   127 lemma ex_has_greatest_nat: "[|P k;  ! y. P y --> m y < b|]  
   128       ==> ? x. P x & (! y. P y --> (m y::nat) <= m x)"
   129 apply (rule ccontr)
   130 apply (cut_tac P = "P" and n = "b - m k" in ex_has_greatest_nat_lemma)
   131 apply (subgoal_tac [3] "m k <= b")
   132 apply auto
   133 done
   134 
   135 lemma GreatestM_nat_lemma: 
   136      "[|P k;  ! y. P y --> m y < b|]  
   137       ==> P (GreatestM m P) & (!y. P y --> (m y::nat) <= m (GreatestM m P))"
   138 apply (unfold GreatestM_def)
   139 apply (rule someI_ex)
   140 apply (erule ex_has_greatest_nat)
   141 apply assumption
   142 done
   143 
   144 lemmas GreatestM_natI = GreatestM_nat_lemma [THEN conjunct1, standard]
   145 
   146 lemma GreatestM_nat_le: "[|P x;  ! y. P y --> m y < b|]  
   147       ==> (m x::nat) <= m (GreatestM m P)"
   148 apply (blast dest: GreatestM_nat_lemma [THEN conjunct2, THEN spec]) 
   149 done
   150 
   151 (** Specialization to GREATEST **)
   152 
   153 lemma GreatestI: 
   154      "[|P (k::nat);  ! y. P y --> y < b|] ==> P (GREATEST x. P x)"
   155 
   156 apply (unfold Greatest_def)
   157 apply (rule GreatestM_natI)
   158 apply auto
   159 done
   160 
   161 lemma Greatest_le: 
   162      "[|P x;  ! y. P y --> y < b|] ==> (x::nat) <= (GREATEST x. P x)"
   163 apply (unfold Greatest_def)
   164 apply (rule GreatestM_nat_le)
   165 apply auto
   166 done
   167 
   168 
   169 ML {*
   170 val LeastMI2 = thm "LeastMI2";
   171 val LeastM_equality = thm "LeastM_equality";
   172 val GreatestM_def = thm "GreatestM_def";
   173 val GreatestMI2 = thm "GreatestMI2";
   174 val GreatestM_equality = thm "GreatestM_equality";
   175 val Greatest_def = thm "Greatest_def";
   176 val Greatest_equality = thm "Greatest_equality";
   177 val GreatestM_natI = thm "GreatestM_natI";
   178 val GreatestM_nat_le = thm "GreatestM_nat_le";
   179 val GreatestI = thm "GreatestI";
   180 val Greatest_le = thm "Greatest_le";
   181 *}
   182 
   183 use "meson_lemmas.ML"
   184 use "Tools/meson.ML"
   185 setup meson_setup
   186 
   187 end