(*  Title:      HOL/Hilbert_Choice.thy

     ID:         $Id$

     Author:     Lawrence C Paulson

     Copyright   2001  University of Cambridge

 *)

 header {* Hilbert's epsilon-operator and everything to do with the Axiom of Choice *}

 theory Hilbert_Choice = NatArith

 files ("Hilbert_Choice_lemmas.ML") ("meson_lemmas.ML") ("Tools/meson.ML"):

 consts

   Eps           :: "('a => bool) => 'a"

 syntax (input)

   "_Eps"        :: "[pttrn, bool] => 'a"                 ("(3\<epsilon>_./ _)" [0, 10] 10)

 syntax (HOL)

   "_Eps"        :: "[pttrn, bool] => 'a"                 ("(3@ _./ _)" [0, 10] 10)

 syntax

   "_Eps"        :: "[pttrn, bool] => 'a"                 ("(3SOME _./ _)" [0, 10] 10)

 translations

   "SOME x. P"             == "Eps (%x. P)"

 axioms  

   someI:        "P (x::'a) ==> P (SOME x. P x)"

 (*used in TFL*)

 lemma tfl_some: "\<forall>P x. P x --> P (Eps P)"

   by (blast intro: someI)

 constdefs  

   inv :: "('a => 'b) => ('b => 'a)"

     "inv(f::'a=>'b) == % y. @x. f(x)=y"

   Inv :: "['a set, 'a => 'b] => ('b => 'a)"

     "Inv A f == (% x. (@ y. y : A & f y = x))"

 use "Hilbert_Choice_lemmas.ML"

 (** Least value operator **)

 constdefs

   LeastM   :: "['a => 'b::ord, 'a => bool] => 'a"

               "LeastM m P == @x. P x & (ALL y. P y --> m x <= m y)"

 syntax

  "@LeastM" :: "[pttrn, 'a=>'b::ord, bool] => 'a" ("LEAST _ WRT _. _" [0,4,10]10)

 translations

                 "LEAST x WRT m. P" == "LeastM m (%x. P)"

 lemma LeastMI2:

   "[| P x; !!y. P y ==> m x <= m y;

            !!x. [| P x; \<forall>y. P y --> m x \<le> m y |] ==> Q x |]

    ==> Q (LeastM m P)";

 apply (unfold LeastM_def)

 apply (rule someI2_ex)

 apply  blast

 apply blast

 done

 lemma LeastM_equality:

  "[| P k; !!x. P x ==> m k <= m x |] ==> m (LEAST x WRT m. P x) = 

      (m k::'a::order)";

 apply (rule LeastMI2)

 apply   assumption

 apply  blast

 apply (blast intro!: order_antisym) 

 done

 lemma wf_linord_ex_has_least:

      "[|wf r;  ALL x y. ((x,y):r^+) = ((y,x)~:r^*);  P k|]  \

 \     ==> EX x. P x & (!y. P y --> (m x,m y):r^*)" 

 apply (drule wf_trancl [THEN wf_eq_minimal [THEN iffD1]])

 apply (drule_tac x = "m`Collect P" in spec)

 apply force

 done

 (* successor of obsolete nonempty_has_least *)

 lemma ex_has_least_nat:

      "P k ==> EX x. P x & (ALL y. P y --> m x <= (m y::nat))"

 apply (simp only: pred_nat_trancl_eq_le [symmetric])

 apply (rule wf_pred_nat [THEN wf_linord_ex_has_least])

 apply (simp (no_asm) add: less_eq not_le_iff_less pred_nat_trancl_eq_le)

 apply assumption

 done

 lemma LeastM_nat_lemma: 

   "P k ==> P (LeastM m P) & (ALL y. P y --> m (LeastM m P) <= (m y::nat))"

 apply (unfold LeastM_def)

 apply (rule someI_ex)

 apply (erule ex_has_least_nat)

 done

 lemmas LeastM_natI = LeastM_nat_lemma [THEN conjunct1, standard]

 lemma LeastM_nat_le: "P x ==> m (LeastM m P) <= (m x::nat)"

 apply (rule LeastM_nat_lemma [THEN conjunct2, THEN spec, THEN mp])

 apply assumption

 apply assumption

 done

 (** Greatest value operator **)

 constdefs

   GreatestM   :: "['a => 'b::ord, 'a => bool] => 'a"

               "GreatestM m P == @x. P x & (ALL y. P y --> m y <= m x)"

   Greatest    :: "('a::ord => bool) => 'a"         (binder "GREATEST " 10)

               "Greatest     == GreatestM (%x. x)"

 syntax

  "@GreatestM" :: "[pttrn, 'a=>'b::ord, bool] => 'a"

                                         ("GREATEST _ WRT _. _" [0,4,10]10)

 translations

               "GREATEST x WRT m. P" == "GreatestM m (%x. P)"

 lemma GreatestMI2:

      "[| P x;

 	 !!y. P y ==> m y <= m x;

          !!x. [| P x; \<forall>y. P y --> m y \<le> m x |] ==> Q x |]

       ==> Q (GreatestM m P)";

 apply (unfold GreatestM_def)

 apply (rule someI2_ex)

 apply  blast

 apply blast

 done

 lemma GreatestM_equality:

  "[| P k;  !!x. P x ==> m x <= m k |]

   ==> m (GREATEST x WRT m. P x) = (m k::'a::order)";

 apply (rule_tac m=m in GreatestMI2)

 apply   assumption

 apply  blast

 apply (blast intro!: order_antisym) 

 done

 lemma Greatest_equality:

   "[| P (k::'a::order); !!x. P x ==> x <= k |] ==> (GREATEST x. P x) = k";

 apply (unfold Greatest_def)

 apply (erule GreatestM_equality)

 apply blast

 done

 lemma ex_has_greatest_nat_lemma:

      "[|P k;  ALL x. P x --> (EX y. P y & ~ ((m y::nat) <= m x))|]  

       ==> EX y. P y & ~ (m y < m k + n)"

 apply (induct_tac "n")

 apply force

 (*ind step*)

 apply (force simp add: le_Suc_eq)

 done

 lemma ex_has_greatest_nat: "[|P k;  ! y. P y --> m y < b|]  

       ==> ? x. P x & (! y. P y --> (m y::nat) <= m x)"

 apply (rule ccontr)

 apply (cut_tac P = "P" and n = "b - m k" in ex_has_greatest_nat_lemma)

 apply (subgoal_tac [3] "m k <= b")

 apply auto

 done

 lemma GreatestM_nat_lemma: 

      "[|P k;  ! y. P y --> m y < b|]  

       ==> P (GreatestM m P) & (!y. P y --> (m y::nat) <= m (GreatestM m P))"

 apply (unfold GreatestM_def)

 apply (rule someI_ex)

 apply (erule ex_has_greatest_nat)

 apply assumption

 done

 lemmas GreatestM_natI = GreatestM_nat_lemma [THEN conjunct1, standard]

 lemma GreatestM_nat_le: "[|P x;  ! y. P y --> m y < b|]  

       ==> (m x::nat) <= m (GreatestM m P)"

 apply (blast dest: GreatestM_nat_lemma [THEN conjunct2, THEN spec]) 

 done

 (** Specialization to GREATEST **)

 lemma GreatestI: 

      "[|P (k::nat);  ! y. P y --> y < b|] ==> P (GREATEST x. P x)"

 apply (unfold Greatest_def)

 apply (rule GreatestM_natI)

 apply auto

 done

 lemma Greatest_le: 

      "[|P x;  ! y. P y --> y < b|] ==> (x::nat) <= (GREATEST x. P x)"

 apply (unfold Greatest_def)

 apply (rule GreatestM_nat_le)

 apply auto

 done

 use "meson_lemmas.ML"

 use "Tools/meson.ML"

 setup meson_setup

 end