expand_fun_eq -> ext_iff
expand_set_eq -> set_ext_iff
Naming in line now with multisets
(* Title: HOL/Hilbert_Choice.thy
Author: Lawrence C Paulson, Tobias Nipkow
Copyright 2001 University of Cambridge
*)
header {* Hilbert's Epsilon-Operator and the Axiom of Choice *}
theory Hilbert_Choice
imports Nat Wellfounded Plain
uses ("Tools/meson.ML")
("Tools/choice_specification.ML")
begin
subsection {* Hilbert's epsilon *}
axiomatization Eps :: "('a => bool) => 'a" where
someI: "P x ==> P (Eps P)"
syntax (epsilon)
"_Eps" :: "[pttrn, bool] => 'a" ("(3\<some>_./ _)" [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" == "CONST Eps (%x. P)"
print_translation {*
[(@{const_syntax Eps}, fn [Abs abs] =>
let val (x, t) = atomic_abs_tr' abs
in Syntax.const @{syntax_const "_Eps"} $ x $ t end)]
*} -- {* to avoid eta-contraction of body *}
definition inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where
"inv_into A f == %x. SOME y. y : A & f y = x"
abbreviation inv :: "('a => 'b) => ('b => 'a)" where
"inv == inv_into UNIV"
subsection {*Hilbert's Epsilon-operator*}
text{*Easier to apply than @{text someI} if the witness comes from an
existential formula*}
lemma someI_ex [elim?]: "\<exists>x. P x ==> P (SOME x. P x)"
apply (erule exE)
apply (erule someI)
done
text{*Easier to apply than @{text someI} because the conclusion has only one
occurrence of @{term P}.*}
lemma someI2: "[| P a; !!x. P x ==> Q x |] ==> Q (SOME x. P x)"
by (blast intro: someI)
text{*Easier to apply than @{text someI2} if the witness comes from an
existential formula*}
lemma someI2_ex: "[| \<exists>a. P a; !!x. P x ==> Q x |] ==> Q (SOME x. P x)"
by (blast intro: someI2)
lemma some_equality [intro]:
"[| P a; !!x. P x ==> x=a |] ==> (SOME x. P x) = a"
by (blast intro: someI2)
lemma some1_equality: "[| EX!x. P x; P a |] ==> (SOME x. P x) = a"
by blast
lemma some_eq_ex: "P (SOME x. P x) = (\<exists>x. P x)"
by (blast intro: someI)
lemma some_eq_trivial [simp]: "(SOME y. y=x) = x"
apply (rule some_equality)
apply (rule refl, assumption)
done
lemma some_sym_eq_trivial [simp]: "(SOME y. x=y) = x"
apply (rule some_equality)
apply (rule refl)
apply (erule sym)
done
subsection{*Axiom of Choice, Proved Using the Description Operator*}
text{*Used in @{text "Tools/meson.ML"}*}
lemma choice: "\<forall>x. \<exists>y. Q x y ==> \<exists>f. \<forall>x. Q x (f x)"
by (fast elim: someI)
lemma bchoice: "\<forall>x\<in>S. \<exists>y. Q x y ==> \<exists>f. \<forall>x\<in>S. Q x (f x)"
by (fast elim: someI)
subsection {*Function Inverse*}
lemma inv_def: "inv f = (%y. SOME x. f x = y)"
by(simp add: inv_into_def)
lemma inv_into_into: "x : f ` A ==> inv_into A f x : A"
apply (simp add: inv_into_def)
apply (fast intro: someI2)
done
lemma inv_id [simp]: "inv id = id"
by (simp add: inv_into_def id_def)
lemma inv_into_f_f [simp]:
"[| inj_on f A; x : A |] ==> inv_into A f (f x) = x"
apply (simp add: inv_into_def inj_on_def)
apply (blast intro: someI2)
done
lemma inv_f_f: "inj f ==> inv f (f x) = x"
by simp
lemma f_inv_into_f: "y : f`A ==> f (inv_into A f y) = y"
apply (simp add: inv_into_def)
apply (fast intro: someI2)
done
lemma inv_into_f_eq: "[| inj_on f A; x : A; f x = y |] ==> inv_into A f y = x"
apply (erule subst)
apply (fast intro: inv_into_f_f)
done
lemma inv_f_eq: "[| inj f; f x = y |] ==> inv f y = x"
by (simp add:inv_into_f_eq)
lemma inj_imp_inv_eq: "[| inj f; ALL x. f(g x) = x |] ==> inv f = g"
by (blast intro: ext inv_into_f_eq)
text{*But is it useful?*}
lemma inj_transfer:
assumes injf: "inj f" and minor: "!!y. y \<in> range(f) ==> P(inv f y)"
shows "P x"
proof -
have "f x \<in> range f" by auto
hence "P(inv f (f x))" by (rule minor)
thus "P x" by (simp add: inv_into_f_f [OF injf])
qed
lemma inj_iff: "(inj f) = (inv f o f = id)"
apply (simp add: o_def ext_iff)
apply (blast intro: inj_on_inverseI inv_into_f_f)
done
lemma inv_o_cancel[simp]: "inj f ==> inv f o f = id"
by (simp add: inj_iff)
lemma o_inv_o_cancel[simp]: "inj f ==> g o inv f o f = g"
by (simp add: o_assoc[symmetric])
lemma inv_into_image_cancel[simp]:
"inj_on f A ==> S <= A ==> inv_into A f ` f ` S = S"
by(fastsimp simp: image_def)
lemma inj_imp_surj_inv: "inj f ==> surj (inv f)"
by (blast intro: surjI inv_into_f_f)
lemma surj_f_inv_f: "surj f ==> f(inv f y) = y"
by (simp add: f_inv_into_f surj_range)
lemma inv_into_injective:
assumes eq: "inv_into A f x = inv_into A f y"
and x: "x: f`A"
and y: "y: f`A"
shows "x=y"
proof -
have "f (inv_into A f x) = f (inv_into A f y)" using eq by simp
thus ?thesis by (simp add: f_inv_into_f x y)
qed
lemma inj_on_inv_into: "B <= f`A ==> inj_on (inv_into A f) B"
by (blast intro: inj_onI dest: inv_into_injective injD)
lemma bij_betw_inv_into: "bij_betw f A B ==> bij_betw (inv_into A f) B A"
by (auto simp add: bij_betw_def inj_on_inv_into)
lemma surj_imp_inj_inv: "surj f ==> inj (inv f)"
by (simp add: inj_on_inv_into surj_range)
lemma surj_iff: "(surj f) = (f o inv f = id)"
apply (simp add: o_def ext_iff)
apply (blast intro: surjI surj_f_inv_f)
done
lemma surj_imp_inv_eq: "[| surj f; \<forall>x. g(f x) = x |] ==> inv f = g"
apply (rule ext)
apply (drule_tac x = "inv f x" in spec)
apply (simp add: surj_f_inv_f)
done
lemma bij_imp_bij_inv: "bij f ==> bij (inv f)"
by (simp add: bij_def inj_imp_surj_inv surj_imp_inj_inv)
lemma inv_equality: "[| !!x. g (f x) = x; !!y. f (g y) = y |] ==> inv f = g"
apply (rule ext)
apply (auto simp add: inv_into_def)
done
lemma inv_inv_eq: "bij f ==> inv (inv f) = f"
apply (rule inv_equality)
apply (auto simp add: bij_def surj_f_inv_f)
done
(** bij(inv f) implies little about f. Consider f::bool=>bool such that
f(True)=f(False)=True. Then it's consistent with axiom someI that
inv f could be any function at all, including the identity function.
If inv f=id then inv f is a bijection, but inj f, surj(f) and
inv(inv f)=f all fail.
**)
lemma inv_into_comp:
"[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==>
inv_into A (f o g) x = (inv_into A g o inv_into (g ` A) f) x"
apply (rule inv_into_f_eq)
apply (fast intro: comp_inj_on)
apply (simp add: inv_into_into)
apply (simp add: f_inv_into_f inv_into_into)
done
lemma o_inv_distrib: "[| bij f; bij g |] ==> inv (f o g) = inv g o inv f"
apply (rule inv_equality)
apply (auto simp add: bij_def surj_f_inv_f)
done
lemma image_surj_f_inv_f: "surj f ==> f ` (inv f ` A) = A"
by (simp add: image_eq_UN surj_f_inv_f)
lemma image_inv_f_f: "inj f ==> (inv f) ` (f ` A) = A"
by (simp add: image_eq_UN)
lemma inv_image_comp: "inj f ==> inv f ` (f`X) = X"
by (auto simp add: image_def)
lemma bij_image_Collect_eq: "bij f ==> f ` Collect P = {y. P (inv f y)}"
apply auto
apply (force simp add: bij_is_inj)
apply (blast intro: bij_is_surj [THEN surj_f_inv_f, symmetric])
done
lemma bij_vimage_eq_inv_image: "bij f ==> f -` A = inv f ` A"
apply (auto simp add: bij_is_surj [THEN surj_f_inv_f])
apply (blast intro: bij_is_inj [THEN inv_into_f_f, symmetric])
done
lemma finite_fun_UNIVD1:
assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
and card: "card (UNIV :: 'b set) \<noteq> Suc 0"
shows "finite (UNIV :: 'a set)"
proof -
from fin have finb: "finite (UNIV :: 'b set)" by (rule finite_fun_UNIVD2)
with card have "card (UNIV :: 'b set) \<ge> Suc (Suc 0)"
by (cases "card (UNIV :: 'b set)") (auto simp add: card_eq_0_iff)
then obtain n where "card (UNIV :: 'b set) = Suc (Suc n)" "n = card (UNIV :: 'b set) - Suc (Suc 0)" by auto
then obtain b1 b2 where b1b2: "(b1 :: 'b) \<noteq> (b2 :: 'b)" by (auto simp add: card_Suc_eq)
from fin have "finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1))" by (rule finite_imageI)
moreover have "UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1)"
proof (rule UNIV_eq_I)
fix x :: 'a
from b1b2 have "x = inv (\<lambda>y. if y = x then b1 else b2) b1" by (simp add: inv_into_def)
thus "x \<in> range (\<lambda>f\<Colon>'a \<Rightarrow> 'b. inv f b1)" by blast
qed
ultimately show "finite (UNIV :: 'a set)" by simp
qed
subsection {*Other Consequences of Hilbert's Epsilon*}
text {*Hilbert's Epsilon and the @{term split} Operator*}
text{*Looping simprule*}
lemma split_paired_Eps: "(SOME x. P x) = (SOME (a,b). P(a,b))"
by simp
lemma Eps_split: "Eps (split P) = (SOME xy. P (fst xy) (snd xy))"
by (simp add: split_def)
lemma Eps_split_eq [simp]: "(@(x',y'). x = x' & y = y') = (x,y)"
by blast
text{*A relation is wellfounded iff it has no infinite descending chain*}
lemma wf_iff_no_infinite_down_chain:
"wf r = (~(\<exists>f. \<forall>i. (f(Suc i),f i) \<in> r))"
apply (simp only: wf_eq_minimal)
apply (rule iffI)
apply (rule notI)
apply (erule exE)
apply (erule_tac x = "{w. \<exists>i. w=f i}" in allE, blast)
apply (erule contrapos_np, simp, clarify)
apply (subgoal_tac "\<forall>n. nat_rec x (%i y. @z. z:Q & (z,y) :r) n \<in> Q")
apply (rule_tac x = "nat_rec x (%i y. @z. z:Q & (z,y) :r)" in exI)
apply (rule allI, simp)
apply (rule someI2_ex, blast, blast)
apply (rule allI)
apply (induct_tac "n", simp_all)
apply (rule someI2_ex, blast+)
done
lemma wf_no_infinite_down_chainE:
assumes "wf r" obtains k where "(f (Suc k), f k) \<notin> r"
using `wf r` wf_iff_no_infinite_down_chain[of r] by blast
text{*A dynamically-scoped fact for TFL *}
lemma tfl_some: "\<forall>P x. P x --> P (Eps P)"
by (blast intro: someI)
subsection {* Least value operator *}
definition
LeastM :: "['a => 'b::ord, 'a => bool] => 'a" where
"LeastM m P == SOME x. P x & (\<forall>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" == "CONST 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 (simp add: LeastM_def)
apply (rule someI2_ex, blast, 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, assumption, blast)
apply (blast intro!: order_antisym)
done
lemma wf_linord_ex_has_least:
"wf r ==> \<forall>x y. ((x,y):r^+) = ((y,x)~:r^*) ==> P k
==> \<exists>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, force)
done
lemma ex_has_least_nat:
"P k ==> \<exists>x. P x & (\<forall>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 linorder_not_le pred_nat_trancl_eq_le, assumption)
done
lemma LeastM_nat_lemma:
"P k ==> P (LeastM m P) & (\<forall>y. P y --> m (LeastM m P) <= (m y::nat))"
apply (simp add: 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)"
by (rule LeastM_nat_lemma [THEN conjunct2, THEN spec, THEN mp], assumption, assumption)
subsection {* Greatest value operator *}
definition
GreatestM :: "['a => 'b::ord, 'a => bool] => 'a" where
"GreatestM m P == SOME x. P x & (\<forall>y. P y --> m y <= m x)"
definition
Greatest :: "('a::ord => bool) => 'a" (binder "GREATEST " 10) where
"Greatest == GreatestM (%x. x)"
syntax
"_GreatestM" :: "[pttrn, 'a => 'b::ord, bool] => 'a"
("GREATEST _ WRT _. _" [0, 4, 10] 10)
translations
"GREATEST x WRT m. P" == "CONST 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 (simp add: GreatestM_def)
apply (rule someI2_ex, blast, 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, assumption, 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 (simp add: Greatest_def)
apply (erule GreatestM_equality, blast)
done
lemma ex_has_greatest_nat_lemma:
"P k ==> \<forall>x. P x --> (\<exists>y. P y & ~ ((m y::nat) <= m x))
==> \<exists>y. P y & ~ (m y < m k + n)"
apply (induct n, force)
apply (force simp add: le_Suc_eq)
done
lemma ex_has_greatest_nat:
"P k ==> \<forall>y. P y --> m y < b
==> \<exists>x. P x & (\<forall>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", auto)
done
lemma GreatestM_nat_lemma:
"P k ==> \<forall>y. P y --> m y < b
==> P (GreatestM m P) & (\<forall>y. P y --> (m y::nat) <= m (GreatestM m P))"
apply (simp add: GreatestM_def)
apply (rule someI_ex)
apply (erule ex_has_greatest_nat, assumption)
done
lemmas GreatestM_natI = GreatestM_nat_lemma [THEN conjunct1, standard]
lemma GreatestM_nat_le:
"P x ==> \<forall>y. P y --> m y < b
==> (m x::nat) <= m (GreatestM m P)"
apply (blast dest: GreatestM_nat_lemma [THEN conjunct2, THEN spec, of P])
done
text {* \medskip Specialization to @{text GREATEST}. *}
lemma GreatestI: "P (k::nat) ==> \<forall>y. P y --> y < b ==> P (GREATEST x. P x)"
apply (simp add: Greatest_def)
apply (rule GreatestM_natI, auto)
done
lemma Greatest_le:
"P x ==> \<forall>y. P y --> y < b ==> (x::nat) <= (GREATEST x. P x)"
apply (simp add: Greatest_def)
apply (rule GreatestM_nat_le, 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. ~(\<forall>x. P(x)) ==> \<exists>x. ~P(x)"
and meson_not_exD: "!!P. ~(\<exists>x. P(x)) ==> \<forall>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 *}
and meson_not_refl_disj_D: "x ~= x | P ==> P"
by fast+
subsubsection {* Pulling out the existential quantifiers *}
text {* Conjunction *}
lemma meson_conj_exD1: "!!P Q. (\<exists>x. P(x)) & Q ==> \<exists>x. P(x) & Q"
and meson_conj_exD2: "!!P Q. P & (\<exists>x. Q(x)) ==> \<exists>x. P & Q(x)"
by fast+
text {* Disjunction *}
lemma meson_disj_exD: "!!P Q. (\<exists>x. P(x)) | (\<exists>x. Q(x)) ==> \<exists>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. (\<exists>x. P(x)) | Q ==> \<exists>x. P(x) | Q"
and meson_disj_exD2: "!!P Q. P | (\<exists>x. Q(x)) ==> \<exists>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+
subsection{*Lemmas for Meson, the Model Elimination Procedure*}
text{* Generation of contrapositives *}
text{*Inserts negated disjunct after removing the negation; P is a literal.
Model elimination requires assuming the negation of every attempted subgoal,
hence the negated disjuncts.*}
lemma make_neg_rule: "~P|Q ==> ((~P==>P) ==> Q)"
by blast
text{*Version for Plaisted's "Postive refinement" of the Meson procedure*}
lemma make_refined_neg_rule: "~P|Q ==> (P ==> Q)"
by blast
text{*@{term P} should be a literal*}
lemma make_pos_rule: "P|Q ==> ((P==>~P) ==> Q)"
by blast
text{*Versions of @{text make_neg_rule} and @{text make_pos_rule} that don't
insert new assumptions, for ordinary resolution.*}
lemmas make_neg_rule' = make_refined_neg_rule
lemma make_pos_rule': "[|P|Q; ~P|] ==> Q"
by blast
text{* Generation of a goal clause -- put away the final literal *}
lemma make_neg_goal: "~P ==> ((~P==>P) ==> False)"
by blast
lemma make_pos_goal: "P ==> ((P==>~P) ==> False)"
by blast
subsubsection{* Lemmas for Forward Proof*}
text{*There is a similarity to congruence rules*}
(*NOTE: could handle conjunctions (faster?) by
nf(th RS conjunct2) RS (nf(th RS conjunct1) RS conjI) *)
lemma conj_forward: "[| P'&Q'; P' ==> P; Q' ==> Q |] ==> P&Q"
by blast
lemma disj_forward: "[| P'|Q'; P' ==> P; Q' ==> Q |] ==> P|Q"
by blast
(*Version of @{text disj_forward} for removal of duplicate literals*)
lemma disj_forward2:
"[| P'|Q'; P' ==> P; [| Q'; P==>False |] ==> Q |] ==> P|Q"
apply blast
done
lemma all_forward: "[| \<forall>x. P'(x); !!x. P'(x) ==> P(x) |] ==> \<forall>x. P(x)"
by blast
lemma ex_forward: "[| \<exists>x. P'(x); !!x. P'(x) ==> P(x) |] ==> \<exists>x. P(x)"
by blast
subsection {* Meson package *}
use "Tools/meson.ML"
setup Meson.setup
subsection {* Specification package -- Hilbertized version *}
lemma exE_some: "[| Ex P ; c == Eps P |] ==> P c"
by (simp only: someI_ex)
use "Tools/choice_specification.ML"
end