src/HOL/Hilbert_Choice.thy

less_induct, wf_induct_rule;

(*  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"):


subsection {* Hilbert's epsilon *}

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)"


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

  Inv :: "'a set => ('a => 'b) => ('b => 'a)"
  "Inv A f == %x. SOME y. y : A & f y = x"


use "Hilbert_Choice_lemmas.ML"
declare someI_ex [elim?];


lemma tfl_some: "\<forall>P x. P x --> P (Eps P)"
  -- {* dynamically-scoped fact for TFL *}
  by (blast intro: someI)


subsection {* Least value operator *}

constdefs
  LeastM :: "['a => 'b::ord, 'a => bool] => 'a"
  "LeastM m P == SOME 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

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 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


subsection {* Greatest value operator *}

constdefs
  GreatestM :: "['a => 'b::ord, 'a => bool] => 'a"
  "GreatestM m P == SOME 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
  apply (force simp add: le_Suc_eq)
  done

lemma ex_has_greatest_nat:
  "P k ==> ALL y. P y --> m y < b
    ==> EX x. P x & (ALL 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 ==> ALL y. P y --> m y < b
    ==> P (GreatestM m P) & (ALL 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 ==> ALL y. P y --> m y < b
    ==> (m x::nat) <= m (GreatestM m P)"
  apply (blast dest: GreatestM_nat_lemma [THEN conjunct2, THEN spec])
  done


text {* \medskip Specialization to @{text GREATEST}. *}

lemma GreatestI: "P (k::nat) ==> ALL 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 ==> ALL y. P y --> y < b ==> (x::nat) <= (GREATEST x. P x)"
  apply (unfold Greatest_def)
  apply (rule GreatestM_nat_le)
   apply auto
  done


subsection {* The Meson proof procedure *}

subsubsection {* Negation Normal Form *}

text {* de Morgan laws *}

lemma meson_not_conjD: "~(P&Q) ==> ~P | ~Q"
  and meson_not_disjD: "~(P|Q) ==> ~P & ~Q"
  and meson_not_notD: "~~P ==> P"
  and meson_not_allD: "!!P. ~(ALL x. P(x)) ==> EX x. ~P(x)"
  and meson_not_exD: "!!P. ~(EX x. P(x)) ==> ALL x. ~P(x)"
  by fast+

text {* Removal of @{text "-->"} and @{text "<->"} (positive and
negative occurrences) *}

lemma meson_imp_to_disjD: "P-->Q ==> ~P | Q"
  and meson_not_impD: "~(P-->Q) ==> P & ~Q"
  and meson_iff_to_disjD: "P=Q ==> (~P | Q) & (~Q | P)"
  and meson_not_iffD: "~(P=Q) ==> (P | Q) & (~P | ~Q)"
    -- {* Much more efficient than @{prop "(P & ~Q) | (Q & ~P)"} for computing CNF *}
  by fast+


subsubsection {* Pulling out the existential quantifiers *}

text {* Conjunction *}

lemma meson_conj_exD1: "!!P Q. (EX x. P(x)) & Q ==> EX x. P(x) & Q"
  and meson_conj_exD2: "!!P Q. P & (EX x. Q(x)) ==> EX x. P & Q(x)"
  by fast+


text {* Disjunction *}

lemma meson_disj_exD: "!!P Q. (EX x. P(x)) | (EX x. Q(x)) ==> EX x. P(x) | Q(x)"
  -- {* DO NOT USE with forall-Skolemization: makes fewer schematic variables!! *}
  -- {* With ex-Skolemization, makes fewer Skolem constants *}
  and meson_disj_exD1: "!!P Q. (EX x. P(x)) | Q ==> EX x. P(x) | Q"
  and meson_disj_exD2: "!!P Q. P | (EX x. Q(x)) ==> EX x. P | Q(x)"
  by fast+


subsubsection {* Generating clauses for the Meson Proof Procedure *}

text {* Disjunctions *}

lemma meson_disj_assoc: "(P|Q)|R ==> P|(Q|R)"
  and meson_disj_comm: "P|Q ==> Q|P"
  and meson_disj_FalseD1: "False|P ==> P"
  and meson_disj_FalseD2: "P|False ==> P"
  by fast+

use "meson_lemmas.ML"
use "Tools/meson.ML"
setup meson_setup

end