(* 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)"
print_translation {*
(* to avoid eta-contraction of body *)
[("Eps", fn [Abs abs] =>
let val (x,t) = atomic_abs_tr' abs
in Syntax.const "_Eps" $ x $ t end)]
*}
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 Inv_mem: "[| f ` A = B; x \<in> B |] ==> Inv A f x \<in> A"
apply (unfold Inv_def)
apply (fast intro: someI2)
done
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