paulson@11451: (*  Title:      HOL/Hilbert_Choice.thy
paulson@11451:     ID:         $Id$
paulson@11451:     Author:     Lawrence C Paulson
paulson@11451:     Copyright   2001  University of Cambridge
wenzelm@12023: *)
paulson@11451: 
wenzelm@12023: header {* Hilbert's epsilon-operator and everything to do with the Axiom of Choice *}
paulson@11451: 
paulson@11451: theory Hilbert_Choice = NatArith
paulson@11451: files ("Hilbert_Choice_lemmas.ML") ("meson_lemmas.ML") ("Tools/meson.ML"):
paulson@11451: 
wenzelm@12298: 
wenzelm@12298: subsection {* Hilbert's epsilon *}
wenzelm@12298: 
paulson@11451: consts
paulson@11451:   Eps           :: "('a => bool) => 'a"
paulson@11451: 
paulson@11451: syntax (input)
wenzelm@12298:   "_Eps"        :: "[pttrn, bool] => 'a"    ("(3\<epsilon>_./ _)" [0, 10] 10)
paulson@11451: syntax (HOL)
wenzelm@12298:   "_Eps"        :: "[pttrn, bool] => 'a"    ("(3@ _./ _)" [0, 10] 10)
paulson@11451: syntax
wenzelm@12298:   "_Eps"        :: "[pttrn, bool] => 'a"    ("(3SOME _./ _)" [0, 10] 10)
paulson@11451: translations
nipkow@13764:   "SOME x. P" == "Eps (%x. P)"
nipkow@13763: 
nipkow@13763: print_translation {*
nipkow@13763: (* to avoid eta-contraction of body *)
nipkow@13763: [("Eps", fn [Abs abs] =>
nipkow@13763:      let val (x,t) = atomic_abs_tr' abs
nipkow@13763:      in Syntax.const "_Eps" $ x $ t end)]
nipkow@13763: *}
paulson@11451: 
wenzelm@12298: axioms
wenzelm@12298:   someI: "P (x::'a) ==> P (SOME x. P x)"
paulson@11451: 
paulson@11451: 
wenzelm@12298: constdefs
wenzelm@12298:   inv :: "('a => 'b) => ('b => 'a)"
wenzelm@12298:   "inv(f :: 'a => 'b) == %y. SOME x. f x = y"
paulson@11454: 
wenzelm@12298:   Inv :: "'a set => ('a => 'b) => ('b => 'a)"
wenzelm@12298:   "Inv A f == %x. SOME y. y : A & f y = x"
paulson@11451: 
paulson@11451: 
paulson@11451: use "Hilbert_Choice_lemmas.ML"
wenzelm@12372: declare someI_ex [elim?];
wenzelm@12372: 
paulson@13585: lemma Inv_mem: "[| f ` A = B;  x \<in> B |] ==> Inv A f x \<in> A"
paulson@13585: apply (unfold Inv_def)
paulson@13585: apply (fast intro: someI2)
paulson@13585: done
paulson@11451: 
wenzelm@12298: lemma tfl_some: "\<forall>P x. P x --> P (Eps P)"
wenzelm@12298:   -- {* dynamically-scoped fact for TFL *}
wenzelm@12298:   by (blast intro: someI)
paulson@11451: 
wenzelm@12298: 
wenzelm@12298: subsection {* Least value operator *}
paulson@11451: 
paulson@11451: constdefs
wenzelm@12298:   LeastM :: "['a => 'b::ord, 'a => bool] => 'a"
wenzelm@12298:   "LeastM m P == SOME x. P x & (ALL y. P y --> m x <= m y)"
paulson@11451: 
paulson@11451: syntax
wenzelm@12298:   "_LeastM" :: "[pttrn, 'a => 'b::ord, bool] => 'a"    ("LEAST _ WRT _. _" [0, 4, 10] 10)
paulson@11451: translations
wenzelm@12298:   "LEAST x WRT m. P" == "LeastM m (%x. P)"
paulson@11451: 
paulson@11451: lemma LeastMI2:
wenzelm@12298:   "P x ==> (!!y. P y ==> m x <= m y)
wenzelm@12298:     ==> (!!x. P x ==> \<forall>y. P y --> m x \<le> m y ==> Q x)
wenzelm@12298:     ==> Q (LeastM m P)"
wenzelm@12298:   apply (unfold LeastM_def)
wenzelm@12298:   apply (rule someI2_ex)
wenzelm@12298:    apply blast
wenzelm@12298:   apply blast
wenzelm@12298:   done
paulson@11451: 
paulson@11451: lemma LeastM_equality:
wenzelm@12298:   "P k ==> (!!x. P x ==> m k <= m x)
wenzelm@12298:     ==> m (LEAST x WRT m. P x) = (m k::'a::order)"
wenzelm@12298:   apply (rule LeastMI2)
wenzelm@12298:     apply assumption
wenzelm@12298:    apply blast
wenzelm@12298:   apply (blast intro!: order_antisym)
wenzelm@12298:   done
paulson@11451: 
paulson@11454: lemma wf_linord_ex_has_least:
wenzelm@12298:   "wf r ==> ALL x y. ((x,y):r^+) = ((y,x)~:r^*) ==> P k
wenzelm@12298:     ==> EX x. P x & (!y. P y --> (m x,m y):r^*)"
wenzelm@12298:   apply (drule wf_trancl [THEN wf_eq_minimal [THEN iffD1]])
wenzelm@12298:   apply (drule_tac x = "m`Collect P" in spec)
wenzelm@12298:   apply force
wenzelm@12298:   done
paulson@11454: 
paulson@11454: lemma ex_has_least_nat:
wenzelm@12298:     "P k ==> EX x. P x & (ALL y. P y --> m x <= (m y::nat))"
wenzelm@12298:   apply (simp only: pred_nat_trancl_eq_le [symmetric])
wenzelm@12298:   apply (rule wf_pred_nat [THEN wf_linord_ex_has_least])
wenzelm@12298:    apply (simp add: less_eq not_le_iff_less pred_nat_trancl_eq_le)
wenzelm@12298:   apply assumption
wenzelm@12298:   done
paulson@11454: 
wenzelm@12298: lemma LeastM_nat_lemma:
wenzelm@12298:     "P k ==> P (LeastM m P) & (ALL y. P y --> m (LeastM m P) <= (m y::nat))"
wenzelm@12298:   apply (unfold LeastM_def)
wenzelm@12298:   apply (rule someI_ex)
wenzelm@12298:   apply (erule ex_has_least_nat)
wenzelm@12298:   done
paulson@11454: 
paulson@11454: lemmas LeastM_natI = LeastM_nat_lemma [THEN conjunct1, standard]
paulson@11454: 
paulson@11454: lemma LeastM_nat_le: "P x ==> m (LeastM m P) <= (m x::nat)"
wenzelm@12298:   apply (rule LeastM_nat_lemma [THEN conjunct2, THEN spec, THEN mp])
wenzelm@12298:    apply assumption
wenzelm@12298:   apply assumption
wenzelm@12298:   done
paulson@11454: 
paulson@11451: 
wenzelm@12298: subsection {* Greatest value operator *}
paulson@11451: 
paulson@11451: constdefs
wenzelm@12298:   GreatestM :: "['a => 'b::ord, 'a => bool] => 'a"
wenzelm@12298:   "GreatestM m P == SOME x. P x & (ALL y. P y --> m y <= m x)"
wenzelm@12298: 
wenzelm@12298:   Greatest :: "('a::ord => bool) => 'a"    (binder "GREATEST " 10)
wenzelm@12298:   "Greatest == GreatestM (%x. x)"
paulson@11451: 
paulson@11451: syntax
wenzelm@12298:   "_GreatestM" :: "[pttrn, 'a=>'b::ord, bool] => 'a"
wenzelm@12298:       ("GREATEST _ WRT _. _" [0, 4, 10] 10)
paulson@11451: 
paulson@11451: translations
wenzelm@12298:   "GREATEST x WRT m. P" == "GreatestM m (%x. P)"
paulson@11451: 
paulson@11451: lemma GreatestMI2:
wenzelm@12298:   "P x ==> (!!y. P y ==> m y <= m x)
wenzelm@12298:     ==> (!!x. P x ==> \<forall>y. P y --> m y \<le> m x ==> Q x)
wenzelm@12298:     ==> Q (GreatestM m P)"
wenzelm@12298:   apply (unfold GreatestM_def)
wenzelm@12298:   apply (rule someI2_ex)
wenzelm@12298:    apply blast
wenzelm@12298:   apply blast
wenzelm@12298:   done
paulson@11451: 
paulson@11451: lemma GreatestM_equality:
wenzelm@12298:  "P k ==> (!!x. P x ==> m x <= m k)
wenzelm@12298:     ==> m (GREATEST x WRT m. P x) = (m k::'a::order)"
wenzelm@12298:   apply (rule_tac m = m in GreatestMI2)
wenzelm@12298:     apply assumption
wenzelm@12298:    apply blast
wenzelm@12298:   apply (blast intro!: order_antisym)
wenzelm@12298:   done
paulson@11451: 
paulson@11451: lemma Greatest_equality:
wenzelm@12298:   "P (k::'a::order) ==> (!!x. P x ==> x <= k) ==> (GREATEST x. P x) = k"
wenzelm@12298:   apply (unfold Greatest_def)
wenzelm@12298:   apply (erule GreatestM_equality)
wenzelm@12298:   apply blast
wenzelm@12298:   done
paulson@11451: 
paulson@11451: lemma ex_has_greatest_nat_lemma:
wenzelm@12298:   "P k ==> ALL x. P x --> (EX y. P y & ~ ((m y::nat) <= m x))
wenzelm@12298:     ==> EX y. P y & ~ (m y < m k + n)"
wenzelm@12298:   apply (induct_tac n)
wenzelm@12298:    apply force
wenzelm@12298:   apply (force simp add: le_Suc_eq)
wenzelm@12298:   done
paulson@11451: 
wenzelm@12298: lemma ex_has_greatest_nat:
wenzelm@12298:   "P k ==> ALL y. P y --> m y < b
wenzelm@12298:     ==> EX x. P x & (ALL y. P y --> (m y::nat) <= m x)"
wenzelm@12298:   apply (rule ccontr)
wenzelm@12298:   apply (cut_tac P = P and n = "b - m k" in ex_has_greatest_nat_lemma)
wenzelm@12298:     apply (subgoal_tac [3] "m k <= b")
wenzelm@12298:      apply auto
wenzelm@12298:   done
paulson@11451: 
wenzelm@12298: lemma GreatestM_nat_lemma:
wenzelm@12298:   "P k ==> ALL y. P y --> m y < b
wenzelm@12298:     ==> P (GreatestM m P) & (ALL y. P y --> (m y::nat) <= m (GreatestM m P))"
wenzelm@12298:   apply (unfold GreatestM_def)
wenzelm@12298:   apply (rule someI_ex)
wenzelm@12298:   apply (erule ex_has_greatest_nat)
wenzelm@12298:   apply assumption
wenzelm@12298:   done
paulson@11451: 
paulson@11451: lemmas GreatestM_natI = GreatestM_nat_lemma [THEN conjunct1, standard]
paulson@11451: 
wenzelm@12298: lemma GreatestM_nat_le:
wenzelm@12298:   "P x ==> ALL y. P y --> m y < b
wenzelm@12298:     ==> (m x::nat) <= m (GreatestM m P)"
wenzelm@12298:   apply (blast dest: GreatestM_nat_lemma [THEN conjunct2, THEN spec])
wenzelm@12298:   done
wenzelm@12298: 
wenzelm@12298: 
wenzelm@12298: text {* \medskip Specialization to @{text GREATEST}. *}
wenzelm@12298: 
wenzelm@12298: lemma GreatestI: "P (k::nat) ==> ALL y. P y --> y < b ==> P (GREATEST x. P x)"
wenzelm@12298:   apply (unfold Greatest_def)
wenzelm@12298:   apply (rule GreatestM_natI)
wenzelm@12298:    apply auto
wenzelm@12298:   done
paulson@11451: 
wenzelm@12298: lemma Greatest_le:
wenzelm@12298:     "P x ==> ALL y. P y --> y < b ==> (x::nat) <= (GREATEST x. P x)"
wenzelm@12298:   apply (unfold Greatest_def)
wenzelm@12298:   apply (rule GreatestM_nat_le)
wenzelm@12298:    apply auto
wenzelm@12298:   done
wenzelm@12298: 
wenzelm@12298: 
wenzelm@12298: subsection {* The Meson proof procedure *}
paulson@11451: 
wenzelm@12298: subsubsection {* Negation Normal Form *}
wenzelm@12298: 
wenzelm@12298: text {* de Morgan laws *}
wenzelm@12298: 
wenzelm@12298: lemma meson_not_conjD: "~(P&Q) ==> ~P | ~Q"
wenzelm@12298:   and meson_not_disjD: "~(P|Q) ==> ~P & ~Q"
wenzelm@12298:   and meson_not_notD: "~~P ==> P"
wenzelm@12298:   and meson_not_allD: "!!P. ~(ALL x. P(x)) ==> EX x. ~P(x)"
wenzelm@12298:   and meson_not_exD: "!!P. ~(EX x. P(x)) ==> ALL x. ~P(x)"
wenzelm@12298:   by fast+
paulson@11451: 
wenzelm@12298: text {* Removal of @{text "-->"} and @{text "<->"} (positive and
wenzelm@12298: negative occurrences) *}
wenzelm@12298: 
wenzelm@12298: lemma meson_imp_to_disjD: "P-->Q ==> ~P | Q"
wenzelm@12298:   and meson_not_impD: "~(P-->Q) ==> P & ~Q"
wenzelm@12298:   and meson_iff_to_disjD: "P=Q ==> (~P | Q) & (~Q | P)"
wenzelm@12298:   and meson_not_iffD: "~(P=Q) ==> (P | Q) & (~P | ~Q)"
wenzelm@12298:     -- {* Much more efficient than @{prop "(P & ~Q) | (Q & ~P)"} for computing CNF *}
wenzelm@12298:   by fast+
wenzelm@12298: 
wenzelm@12298: 
wenzelm@12298: subsubsection {* Pulling out the existential quantifiers *}
wenzelm@12298: 
wenzelm@12298: text {* Conjunction *}
wenzelm@12298: 
wenzelm@12298: lemma meson_conj_exD1: "!!P Q. (EX x. P(x)) & Q ==> EX x. P(x) & Q"
wenzelm@12298:   and meson_conj_exD2: "!!P Q. P & (EX x. Q(x)) ==> EX x. P & Q(x)"
wenzelm@12298:   by fast+
wenzelm@12298: 
paulson@11451: 
wenzelm@12298: text {* Disjunction *}
wenzelm@12298: 
wenzelm@12298: lemma meson_disj_exD: "!!P Q. (EX x. P(x)) | (EX x. Q(x)) ==> EX x. P(x) | Q(x)"
wenzelm@12298:   -- {* DO NOT USE with forall-Skolemization: makes fewer schematic variables!! *}
wenzelm@12298:   -- {* With ex-Skolemization, makes fewer Skolem constants *}
wenzelm@12298:   and meson_disj_exD1: "!!P Q. (EX x. P(x)) | Q ==> EX x. P(x) | Q"
wenzelm@12298:   and meson_disj_exD2: "!!P Q. P | (EX x. Q(x)) ==> EX x. P | Q(x)"
wenzelm@12298:   by fast+
wenzelm@12298: 
paulson@11451: 
wenzelm@12298: subsubsection {* Generating clauses for the Meson Proof Procedure *}
wenzelm@12298: 
wenzelm@12298: text {* Disjunctions *}
wenzelm@12298: 
wenzelm@12298: lemma meson_disj_assoc: "(P|Q)|R ==> P|(Q|R)"
wenzelm@12298:   and meson_disj_comm: "P|Q ==> Q|P"
wenzelm@12298:   and meson_disj_FalseD1: "False|P ==> P"
wenzelm@12298:   and meson_disj_FalseD2: "P|False ==> P"
wenzelm@12298:   by fast+
paulson@11451: 
paulson@11451: use "meson_lemmas.ML"
paulson@11451: use "Tools/meson.ML"
paulson@11451: setup meson_setup
paulson@11451: 
paulson@11451: end