(* Title: HOL/Hilbert_Choice.thy ID: $Id$ Author: Lawrence C Paulson Copyright 2001 University of Cambridge *) header {* Hilbert's Epsilon-Operator and the Axiom of Choice *} theory Hilbert_Choice = NatArith files ("Tools/meson.ML") ("Tools/specification_package.ML"): subsection {* Hilbert's epsilon *} consts Eps :: "('a => bool) => 'a" syntax (epsilon) "_Eps" :: "[pttrn, bool] => 'a" ("(3\_./ _)" [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" subsection {*Hilbert's Epsilon-operator*} text{*Easier to apply than @{text someI} if the witness comes from an existential formula*} lemma someI_ex [elim?]: "\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: "[| \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 intro: some_equality) lemma some_eq_ex: "P (SOME x. P x) = (\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: "\x. \y. Q x y ==> \f. \x. Q x (f x)" by (fast elim: someI) lemma bchoice: "\x\S. \y. Q x y ==> \f. \x\S. Q x (f x)" by (fast elim: someI) subsection {*Function Inverse*} lemma inv_id [simp]: "inv id = id" by (simp add: inv_def id_def) text{*A one-to-one function has an inverse.*} lemma inv_f_f [simp]: "inj f ==> inv f (f x) = x" by (simp add: inv_def inj_eq) lemma inv_f_eq: "[| inj f; f x = y |] ==> inv f y = x" apply (erule subst) apply (erule inv_f_f) done lemma inj_imp_inv_eq: "[| inj f; \x. f(g x) = x |] ==> inv f = g" by (blast intro: ext inv_f_eq) text{*But is it useful?*} lemma inj_transfer: assumes injf: "inj f" and minor: "!!y. y \ range(f) ==> P(inv f y)" shows "P x" proof - have "f x \ range f" by auto hence "P(inv f (f x))" by (rule minor) thus "P x" by (simp add: inv_f_f [OF injf]) qed lemma inj_iff: "(inj f) = (inv f o f = id)" apply (simp add: o_def expand_fun_eq) apply (blast intro: inj_on_inverseI inv_f_f) done lemma inj_imp_surj_inv: "inj f ==> surj (inv f)" by (blast intro: surjI inv_f_f) lemma f_inv_f: "y \ range(f) ==> f(inv f y) = y" apply (simp add: inv_def) apply (fast intro: someI) done lemma surj_f_inv_f: "surj f ==> f(inv f y) = y" by (simp add: f_inv_f surj_range) lemma inv_injective: assumes eq: "inv f x = inv f y" and x: "x: range f" and y: "y: range f" shows "x=y" proof - have "f (inv f x) = f (inv f y)" using eq by simp thus ?thesis by (simp add: f_inv_f x y) qed lemma inj_on_inv: "A <= range(f) ==> inj_on (inv f) A" by (fast intro: inj_onI elim: inv_injective injD) lemma surj_imp_inj_inv: "surj f ==> inj (inv f)" by (simp add: inj_on_inv surj_range) lemma surj_iff: "(surj f) = (f o inv f = id)" apply (simp add: o_def expand_fun_eq) apply (blast intro: surjI surj_f_inv_f) done lemma surj_imp_inv_eq: "[| surj f; \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_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 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_f_f, symmetric]) done subsection {*Inverse of a PI-function (restricted domain)*} lemma Inv_f_f: "[| inj_on f A; x \ A |] ==> Inv A f (f x) = x" apply (simp add: Inv_def inj_on_def) apply (blast intro: someI2) done lemma f_Inv_f: "y \ f`A ==> f (Inv A f y) = y" apply (simp add: Inv_def) apply (fast intro: someI2) done lemma Inv_injective: assumes eq: "Inv A f x = Inv A f y" and x: "x: f`A" and y: "y: f`A" shows "x=y" proof - have "f (Inv A f x) = f (Inv A f y)" using eq by simp thus ?thesis by (simp add: f_Inv_f x y) qed lemma inj_on_Inv: "B <= f`A ==> inj_on (Inv A f) B" apply (rule inj_onI) apply (blast intro: inj_onI dest: Inv_injective injD) done lemma Inv_mem: "[| f ` A = B; x \ B |] ==> Inv A f x \ A" apply (simp add: Inv_def) apply (fast intro: someI2) done lemma Inv_f_eq: "[| inj_on f A; f x = y; x \ A |] ==> Inv A f y = x" apply (erule subst) apply (erule Inv_f_f, assumption) done lemma Inv_comp: "[| inj_on f (g ` A); inj_on g A; x \ f ` g ` A |] ==> Inv A (f o g) x = (Inv A g o Inv (g ` A) f) x" apply simp apply (rule Inv_f_eq) apply (fast intro: comp_inj_on) apply (simp add: f_Inv_f Inv_mem) apply (simp add: Inv_mem) done 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 add: split_Pair_apply) 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 = (~(\f. \i. (f(Suc i),f i) \ r))" apply (simp only: wf_eq_minimal) apply (rule iffI) apply (rule notI) apply (erule exE) apply (erule_tac x = "{w. \i. w=f i}" in allE, blast) apply (erule contrapos_np, simp, clarify) apply (subgoal_tac "\n. nat_rec x (%i y. @z. z:Q & (z,y) :r) n \ 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 text{*A dynamically-scoped fact for TFL *} lemma tfl_some: "\P x. P x --> P (Eps P)" by (blast intro: someI) subsection {* Least value operator *} constdefs LeastM :: "['a => 'b::ord, 'a => bool] => 'a" "LeastM m P == SOME x. P x & (\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 ==> \y. P y --> m x \ 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 ==> \x y. ((x,y):r^+) = ((y,x)~:r^*) ==> P k ==> \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 ==> \x. P x & (\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, assumption) done lemma LeastM_nat_lemma: "P k ==> P (LeastM m P) & (\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 *} constdefs GreatestM :: "['a => 'b::ord, 'a => bool] => 'a" "GreatestM m P == SOME x. P x & (\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 ==> \y. P y --> m y \ 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 ==> \x. P x --> (\y. P y & ~ ((m y::nat) <= m x)) ==> \y. P y & ~ (m y < m k + n)" apply (induct_tac n, force) 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", 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 (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 ==> \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) ==> \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 ==> \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. ~(\x. P(x)) ==> \x. ~P(x)" and meson_not_exD: "!!P. ~(\x. P(x)) ==> \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. (\x. P(x)) & Q ==> \x. P(x) & Q" and meson_conj_exD2: "!!P Q. P & (\x. Q(x)) ==> \x. P & Q(x)" by fast+ text {* Disjunction *} lemma meson_disj_exD: "!!P Q. (\x. P(x)) | (\x. Q(x)) ==> \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. (\x. P(x)) | Q ==> \x. P(x) | Q" and meson_disj_exD2: "!!P Q. P | (\x. Q(x)) ==> \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: "[| \x. P'(x); !!x. P'(x) ==> P(x) |] ==> \x. P(x)" by blast lemma ex_forward: "[| \x. P'(x); !!x. P'(x) ==> P(x) |] ==> \x. P(x)" by blast ML {* val inv_def = thm "inv_def"; val Inv_def = thm "Inv_def"; val someI = thm "someI"; val someI_ex = thm "someI_ex"; val someI2 = thm "someI2"; val someI2_ex = thm "someI2_ex"; val some_equality = thm "some_equality"; val some1_equality = thm "some1_equality"; val some_eq_ex = thm "some_eq_ex"; val some_eq_trivial = thm "some_eq_trivial"; val some_sym_eq_trivial = thm "some_sym_eq_trivial"; val choice = thm "choice"; val bchoice = thm "bchoice"; val inv_id = thm "inv_id"; val inv_f_f = thm "inv_f_f"; val inv_f_eq = thm "inv_f_eq"; val inj_imp_inv_eq = thm "inj_imp_inv_eq"; val inj_transfer = thm "inj_transfer"; val inj_iff = thm "inj_iff"; val inj_imp_surj_inv = thm "inj_imp_surj_inv"; val f_inv_f = thm "f_inv_f"; val surj_f_inv_f = thm "surj_f_inv_f"; val inv_injective = thm "inv_injective"; val inj_on_inv = thm "inj_on_inv"; val surj_imp_inj_inv = thm "surj_imp_inj_inv"; val surj_iff = thm "surj_iff"; val surj_imp_inv_eq = thm "surj_imp_inv_eq"; val bij_imp_bij_inv = thm "bij_imp_bij_inv"; val inv_equality = thm "inv_equality"; val inv_inv_eq = thm "inv_inv_eq"; val o_inv_distrib = thm "o_inv_distrib"; val image_surj_f_inv_f = thm "image_surj_f_inv_f"; val image_inv_f_f = thm "image_inv_f_f"; val inv_image_comp = thm "inv_image_comp"; val bij_image_Collect_eq = thm "bij_image_Collect_eq"; val bij_vimage_eq_inv_image = thm "bij_vimage_eq_inv_image"; val Inv_f_f = thm "Inv_f_f"; val f_Inv_f = thm "f_Inv_f"; val Inv_injective = thm "Inv_injective"; val inj_on_Inv = thm "inj_on_Inv"; val split_paired_Eps = thm "split_paired_Eps"; val Eps_split = thm "Eps_split"; val Eps_split_eq = thm "Eps_split_eq"; val wf_iff_no_infinite_down_chain = thm "wf_iff_no_infinite_down_chain"; val Inv_mem = thm "Inv_mem"; val Inv_f_eq = thm "Inv_f_eq"; val Inv_comp = thm "Inv_comp"; val tfl_some = thm "tfl_some"; val make_neg_rule = thm "make_neg_rule"; val make_refined_neg_rule = thm "make_refined_neg_rule"; val make_pos_rule = thm "make_pos_rule"; val make_neg_rule' = thm "make_neg_rule'"; val make_pos_rule' = thm "make_pos_rule'"; val make_neg_goal = thm "make_neg_goal"; val make_pos_goal = thm "make_pos_goal"; val conj_forward = thm "conj_forward"; val disj_forward = thm "disj_forward"; val disj_forward2 = thm "disj_forward2"; val all_forward = thm "all_forward"; val ex_forward = thm "ex_forward"; *} use "Tools/meson.ML" setup meson_setup use "Tools/specification_package.ML" end