# HG changeset patch # User paulson # Date 1084958987 -7200 # Node ID a08e916f4946795b8eacb7f2d14f5b14c2a43267 # Parent c90bed2d5bdf371e77c9e3bd9a275a69eb7f414d conversion of Hilbert_Choice to Isar script diff -r c90bed2d5bdf -r a08e916f4946 src/HOL/Hilbert_Choice.thy --- a/src/HOL/Hilbert_Choice.thy Wed May 19 11:24:54 2004 +0200 +++ b/src/HOL/Hilbert_Choice.thy Wed May 19 11:29:47 2004 +0200 @@ -1,13 +1,13 @@ (* Title: HOL/Hilbert_Choice.thy - ID: $Id$ + 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 *} +header {* Hilbert's Epsilon-Operator and the Axiom of Choice *} theory Hilbert_Choice = NatArith -files ("Hilbert_Choice_lemmas.ML") ("meson_lemmas.ML") ("Tools/meson.ML") ("Tools/specification_package.ML"): +files ("Tools/meson.ML") ("Tools/specification_package.ML"): subsection {* Hilbert's epsilon *} @@ -40,26 +40,217 @@ "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" + "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 -use "Hilbert_Choice_lemmas.ML" -declare someI_ex [elim?]; +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_mem: "[| f ` A = B; x \ B |] ==> Inv A f x \ A" -apply (unfold Inv_def) +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_f_eq: - "[| inj_on f A; f x = y; x : A |] ==> Inv A f y = x" +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) - apply assumption + apply (erule Inv_f_f, assumption) done lemma Inv_comp: - "[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==> + "[| 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) @@ -68,8 +259,42 @@ 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)" - -- {* dynamically-scoped fact for TFL *} by (blast intro: someI) @@ -77,7 +302,7 @@ constdefs LeastM :: "['a => 'b::ord, 'a => bool] => 'a" - "LeastM m P == SOME x. P x & (ALL y. P y --> m x <= m y)" + "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) @@ -88,7 +313,7 @@ "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 (unfold LeastM_def) + apply (simp add: LeastM_def) apply (rule someI2_ex, blast, blast) done @@ -100,22 +325,22 @@ 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^*)" + "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 ==> EX x. P x & (ALL y. P y --> m x <= (m y::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) & (ALL y. P y --> m (LeastM m P) <= (m y::nat))" - apply (unfold LeastM_def) + "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 @@ -130,7 +355,7 @@ constdefs GreatestM :: "['a => 'b::ord, 'a => bool] => 'a" - "GreatestM m P == SOME x. P x & (ALL y. P y --> m y <= m x)" + "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)" @@ -146,7 +371,7 @@ "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 (unfold GreatestM_def) + apply (simp add: GreatestM_def) apply (rule someI2_ex, blast, blast) done @@ -159,29 +384,29 @@ lemma Greatest_equality: "P (k::'a::order) ==> (!!x. P x ==> x <= k) ==> (GREATEST x. P x) = k" - apply (unfold Greatest_def) + apply (simp add: Greatest_def) apply (erule GreatestM_equality, 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)" + "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 ==> ALL y. P y --> m y < b - ==> EX x. P x & (ALL y. P y --> (m y::nat) <= m x)" + "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 ==> 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) + "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 @@ -189,7 +414,7 @@ lemmas GreatestM_natI = GreatestM_nat_lemma [THEN conjunct1, standard] lemma GreatestM_nat_le: - "P x ==> ALL y. P y --> m y < b + "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 @@ -197,14 +422,14 @@ 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) +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 ==> ALL y. P y --> y < b ==> (x::nat) <= (GREATEST x. P x)" - apply (unfold Greatest_def) + "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 @@ -218,8 +443,8 @@ 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)" + 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 @@ -237,18 +462,18 @@ 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)" +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. (EX x. P(x)) | (EX x. Q(x)) ==> EX x. P(x) | Q(x)" +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. (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)" + 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+ @@ -262,7 +487,133 @@ and meson_disj_FalseD2: "P|False ==> P" by fast+ -use "meson_lemmas.ML" + +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 diff -r c90bed2d5bdf -r a08e916f4946 src/HOL/Hilbert_Choice_lemmas.ML --- a/src/HOL/Hilbert_Choice_lemmas.ML Wed May 19 11:24:54 2004 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,281 +0,0 @@ -(* Title: HOL/Hilbert_Choice_lemmas - ID: $Id$ - Author: Lawrence C Paulson - Copyright 2001 University of Cambridge - -Lemmas for Hilbert's epsilon-operator and the Axiom of Choice -*) - - -(* ML bindings *) -val someI = thm "someI"; - -section "SOME: Hilbert's Epsilon-operator"; - -(*Easier to apply than someI if witness ?a comes from an EX-formula*) -Goal "EX x. P x ==> P (SOME x. P x)"; -by (etac exE 1); -by (etac someI 1); -qed "someI_ex"; - -(*Easier to apply than someI: conclusion has only one occurrence of P*) -val prems = Goal "[| P a; !!x. P x ==> Q x |] ==> Q (SOME x. P x)"; -by (resolve_tac prems 1); -by (rtac someI 1); -by (resolve_tac prems 1) ; -qed "someI2"; - -(*Easier to apply than someI2 if witness ?a comes from an EX-formula*) -val [major,minor] = Goal "[| EX a. P a; !!x. P x ==> Q x |] ==> Q (SOME x. P x)"; -by (rtac (major RS exE) 1); -by (etac someI2 1 THEN etac minor 1); -qed "someI2_ex"; - -val prems = Goal "[| P a; !!x. P x ==> x=a |] ==> (SOME x. P x) = a"; -by (rtac someI2 1); -by (REPEAT (ares_tac prems 1)) ; -qed "some_equality"; -AddIs [some_equality]; - -Goal "[| EX!x. P x; P a |] ==> (SOME x. P x) = a"; -by (rtac some_equality 1); -by (atac 1); -by (etac ex1E 1); -by (etac all_dupE 1); -by (dtac mp 1); -by (atac 1); -by (etac ssubst 1); -by (etac allE 1); -by (etac mp 1); -by (atac 1); -qed "some1_equality"; - -Goal "P (SOME x. P x) = (EX x. P x)"; -by (rtac iffI 1); -by (etac exI 1); -by (etac exE 1); -by (etac someI 1); -qed "some_eq_ex"; - -Goal "(SOME y. y=x) = x"; -by (rtac some_equality 1); -by (rtac refl 1); -by (atac 1); -qed "some_eq_trivial"; - -Goal "(SOME y. x=y) = x"; -by (rtac some_equality 1); -by (rtac refl 1); -by (etac sym 1); -qed "some_sym_eq_trivial"; -Addsimps [some_eq_trivial, some_sym_eq_trivial]; - - -(** "Axiom" of Choice, proved using the description operator **) - -(*Used in Tools/meson.ML*) -Goal "ALL x. EX y. Q x y ==> EX f. ALL x. Q x (f x)"; -by (fast_tac (claset() addEs [someI]) 1); -qed "choice"; - -Goal "ALL x:S. EX y. Q x y ==> EX f. ALL x:S. Q x (f x)"; -by (fast_tac (claset() addEs [someI]) 1); -qed "bchoice"; - - -section "Function Inverse"; - -val inv_def = thm "inv_def"; -val Inv_def = thm "Inv_def"; - -Goal "inv id = id"; -by (simp_tac (simpset() addsimps [inv_def,id_def]) 1); -qed "inv_id"; -Addsimps [inv_id]; - -(*A one-to-one function has an inverse.*) -Goalw [inv_def] "inj(f) ==> inv f (f x) = x"; -by (asm_simp_tac (simpset() addsimps [inj_eq]) 1); -qed "inv_f_f"; -Addsimps [inv_f_f]; - -Goal "[| inj(f); f x = y |] ==> inv f y = x"; -by (etac subst 1); -by (etac inv_f_f 1); -qed "inv_f_eq"; - -Goal "[| inj f; ALL x. f(g x) = x |] ==> inv f = g"; -by (blast_tac (claset() addIs [ext, inv_f_eq]) 1); -qed "inj_imp_inv_eq"; - -(* Useful??? *) -val [oneone,minor] = Goal - "[| inj(f); !!y. y: range(f) ==> P(inv f y) |] ==> P(x)"; -by (res_inst_tac [("t", "x")] (oneone RS (inv_f_f RS subst)) 1); -by (rtac (rangeI RS minor) 1); -qed "inj_transfer"; - -Goal "(inj f) = (inv f o f = id)"; -by (asm_simp_tac (simpset() addsimps [o_def, expand_fun_eq]) 1); -by (blast_tac (claset() addIs [inj_inverseI, inv_f_f]) 1); -qed "inj_iff"; - -Goal "inj f ==> surj (inv f)"; -by (blast_tac (claset() addIs [surjI, inv_f_f]) 1); -qed "inj_imp_surj_inv"; - -Goalw [inv_def] "y : range(f) ==> f(inv f y) = y"; -by (fast_tac (claset() addIs [someI]) 1); -qed "f_inv_f"; - -Goal "surj f ==> f(inv f y) = y"; -by (asm_simp_tac (simpset() addsimps [f_inv_f, surj_range]) 1); -qed "surj_f_inv_f"; - -Goal "[| inv f x = inv f y; x: range(f); y: range(f) |] ==> x=y"; -by (rtac (arg_cong RS box_equals) 1); -by (REPEAT (ares_tac [f_inv_f] 1)); -qed "inv_injective"; - -Goal "A <= range(f) ==> inj_on (inv f) A"; -by (fast_tac (claset() addIs [inj_onI] - addEs [inv_injective, injD]) 1); -qed "inj_on_inv"; - -Goal "surj f ==> inj (inv f)"; -by (asm_simp_tac (simpset() addsimps [inj_on_inv, surj_range]) 1); -qed "surj_imp_inj_inv"; - -Goal "(surj f) = (f o inv f = id)"; -by (asm_simp_tac (simpset() addsimps [o_def, expand_fun_eq]) 1); -by (blast_tac (claset() addIs [surjI, surj_f_inv_f]) 1); -qed "surj_iff"; - -Goal "[| surj f; ALL x. g(f x) = x |] ==> inv f = g"; -by (rtac ext 1); -by (dres_inst_tac [("x","inv f x")] spec 1); -by (asm_full_simp_tac (simpset() addsimps [surj_f_inv_f]) 1); -qed "surj_imp_inv_eq"; - -Goalw [bij_def] "bij f ==> bij (inv f)"; -by (asm_simp_tac (simpset() addsimps [inj_imp_surj_inv, surj_imp_inj_inv]) 1); -qed "bij_imp_bij_inv"; - -val prems = -Goalw [inv_def] "[| !! x. g (f x) = x; !! y. f (g y) = y |] ==> inv f = g"; -by (rtac ext 1); -by (auto_tac (claset(), simpset() addsimps prems)); -qed "inv_equality"; - -Goalw [bij_def] "bij f ==> inv (inv f) = f"; -by (rtac inv_equality 1); -by (auto_tac (claset(), simpset() addsimps [surj_f_inv_f])); -qed "inv_inv_eq"; - -(** 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. -**) - -Goalw [bij_def] "[| bij f; bij g |] ==> inv (f o g) = inv g o inv f"; -by (rtac (inv_equality) 1); -by (auto_tac (claset(), simpset() addsimps [surj_f_inv_f])); -qed "o_inv_distrib"; - - -Goal "surj f ==> f ` (inv f ` A) = A"; -by (asm_simp_tac (simpset() addsimps [image_eq_UN, surj_f_inv_f]) 1); -qed "image_surj_f_inv_f"; - -Goal "inj f ==> (inv f) ` (f ` A) = A"; -by (asm_simp_tac (simpset() addsimps [image_eq_UN]) 1); -qed "image_inv_f_f"; - -Goalw [image_def] "inj(f) ==> inv(f)`(f`X) = X"; -by Auto_tac; -qed "inv_image_comp"; - -Goal "bij f ==> f ` Collect P = {y. P (inv f y)}"; -by Auto_tac; -by (force_tac (claset(), simpset() addsimps [bij_is_inj]) 1); -by (blast_tac (claset() addIs [bij_is_surj RS surj_f_inv_f RS sym]) 1); -qed "bij_image_Collect_eq"; - -Goal "bij f ==> f -` A = inv f ` A"; -by Safe_tac; -by (asm_simp_tac (simpset() addsimps [bij_is_surj RS surj_f_inv_f]) 2); -by (blast_tac (claset() addIs [bij_is_inj RS inv_f_f RS sym]) 1); -qed "bij_vimage_eq_inv_image"; - - -section "Inverse of a PI-function (restricted domain)"; - -Goal "[| inj_on f A; x : A |] ==> Inv A f (f x) = x"; -by (asm_full_simp_tac (simpset() addsimps [Inv_def, inj_on_def]) 1); -by (blast_tac (claset() addIs [someI2]) 1); -qed "Inv_f_f"; - -Goal "y : f`A ==> f (Inv A f y) = y"; -by (asm_simp_tac (simpset() addsimps [Inv_def]) 1); -by (fast_tac (claset() addIs [someI2]) 1); -qed "f_Inv_f"; - -Goal "[| Inv A f x = Inv A f y; x : f`A; y : f`A |] ==> x=y"; -by (rtac (arg_cong RS box_equals) 1); -by (REPEAT (ares_tac [f_Inv_f] 1)); -qed "Inv_injective"; - -Goal "B <= f`A ==> inj_on (Inv A f) B"; -by (rtac inj_onI 1); -by (blast_tac (claset() addIs [inj_onI] addDs [Inv_injective, injD]) 1); -qed "inj_on_Inv"; - - - -section "split and SOME"; - -(*Can't be added to simpset: loops!*) -Goal "(SOME x. P x) = (SOME (a,b). P(a,b))"; -by (simp_tac (simpset() addsimps [split_Pair_apply]) 1); -qed "split_paired_Eps"; - -Goalw [split_def] "Eps (split P) = (SOME xy. P (fst xy) (snd xy))"; -by (rtac refl 1); -qed "Eps_split"; - -Goal "(@(x',y'). x = x' & y = y') = (x,y)"; -by (Blast_tac 1); -qed "Eps_split_eq"; -Addsimps [Eps_split_eq]; - - -section "A relation is wellfounded iff it has no infinite descending chain"; - -Goalw [wf_eq_minimal RS eq_reflection] - "wf r = (~(EX f. ALL i. (f(Suc i),f i) : r))"; -by (rtac iffI 1); - by (rtac notI 1); - by (etac exE 1); - by (eres_inst_tac [("x","{w. EX i. w=f i}")] allE 1); - by (Blast_tac 1); -by (etac contrapos_np 1); -by (Asm_full_simp_tac 1); -by (Clarify_tac 1); -by (subgoal_tac "ALL n. nat_rec x (%i y. @z. z:Q & (z,y):r) n : Q" 1); - by (res_inst_tac[("x","nat_rec x (%i y. @z. z:Q & (z,y):r)")]exI 1); - by (rtac allI 1); - by (Simp_tac 1); - by (rtac someI2_ex 1); - by (Blast_tac 1); - by (Blast_tac 1); -by (rtac allI 1); -by (induct_tac "n" 1); - by (Asm_simp_tac 1); -by (Simp_tac 1); -by (rtac someI2_ex 1); - by (Blast_tac 1); -by (Blast_tac 1); -qed "wf_iff_no_infinite_down_chain"; - diff -r c90bed2d5bdf -r a08e916f4946 src/HOL/IsaMakefile --- a/src/HOL/IsaMakefile Wed May 19 11:24:54 2004 +0200 +++ b/src/HOL/IsaMakefile Wed May 19 11:29:47 2004 +0200 @@ -82,8 +82,7 @@ $(SRC)/TFL/usyntax.ML $(SRC)/TFL/utils.ML \ Datatype.thy Datatype_Universe.ML Datatype_Universe.thy \ Divides.thy Extraction.thy Finite_Set.ML Finite_Set.thy \ - Fun.thy Gfp.ML Gfp.thy \ - Hilbert_Choice.thy Hilbert_Choice_lemmas.ML HOL.ML \ + Fun.thy Gfp.ML Gfp.thy Hilbert_Choice.thy HOL.ML \ HOL.thy HOL_lemmas.ML Inductive.thy Infinite_Set.thy Integ/Bin.thy \ Integ/cooper_dec.ML Integ/cooper_proof.ML \ Integ/Equiv.thy Integ/IntArith.thy Integ/IntDef.thy \ @@ -114,7 +113,7 @@ Transitive_Closure.thy Transitive_Closure.ML Typedef.thy \ Wellfounded_Recursion.ML Wellfounded_Recursion.thy Wellfounded_Relations.ML \ Wellfounded_Relations.thy arith_data.ML blastdata.ML cladata.ML \ - document/root.tex hologic.ML meson_lemmas.ML simpdata.ML thy_syntax.ML + document/root.tex hologic.ML simpdata.ML thy_syntax.ML @$(ISATOOL) usedir -b -g true $(HOL_PROOF_OBJECTS) $(OUT)/Pure HOL diff -r c90bed2d5bdf -r a08e916f4946 src/HOL/meson_lemmas.ML --- a/src/HOL/meson_lemmas.ML Wed May 19 11:24:54 2004 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,95 +0,0 @@ -(* Title: HOL/meson_lemmas.ML - ID: $Id$ - Author: Lawrence C Paulson, Cambridge University Computer Laboratory - Copyright 1992 University of Cambridge - -Lemmas for Meson. -*) - -(* Generation of contrapositives *) - -(*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.*) -val [major,minor] = Goal "~P|Q ==> ((~P==>P) ==> Q)"; -by (rtac (major RS disjE) 1); -by (rtac notE 1); -by (etac minor 2); -by (ALLGOALS assume_tac); -qed "make_neg_rule"; - -(*For Plaisted's "Postive refinement" of the MESON procedure*) -Goal "~P|Q ==> (P ==> Q)"; -by (Blast_tac 1); -qed "make_refined_neg_rule"; - -(*P should be a literal*) -val [major,minor] = Goal "P|Q ==> ((P==>~P) ==> Q)"; -by (rtac (major RS disjE) 1); -by (rtac notE 1); -by (etac minor 1); -by (ALLGOALS assume_tac); -qed "make_pos_rule"; - -(** Versions of make_neg_rule and make_pos_rule that don't insert new - assumptions, for ordinary resolution. **) - -val make_neg_rule' = make_refined_neg_rule; - -Goal "[|P|Q; ~P|] ==> Q"; -by (Blast_tac 1); -qed "make_pos_rule'"; - -(* Generation of a goal clause -- put away the final literal *) - -val [major,minor] = Goal "~P ==> ((~P==>P) ==> False)"; -by (rtac notE 1); -by (rtac minor 2); -by (ALLGOALS (rtac major)); -qed "make_neg_goal"; - -val [major,minor] = Goal "P ==> ((P==>~P) ==> False)"; -by (rtac notE 1); -by (rtac minor 1); -by (ALLGOALS (rtac major)); -qed "make_pos_goal"; - - -(* Lemmas for forward proof (like congruence rules) *) - -(*NOTE: could handle conjunctions (faster?) by - nf(th RS conjunct2) RS (nf(th RS conjunct1) RS conjI) *) -val major::prems = Goal - "[| P'&Q'; P' ==> P; Q' ==> Q |] ==> P&Q"; -by (rtac (major RS conjE) 1); -by (rtac conjI 1); -by (ALLGOALS (eresolve_tac prems)); -qed "conj_forward"; - -val major::prems = Goal - "[| P'|Q'; P' ==> P; Q' ==> Q |] ==> P|Q"; -by (rtac (major RS disjE) 1); -by (ALLGOALS (dresolve_tac prems)); -by (ALLGOALS (eresolve_tac [disjI1,disjI2])); -qed "disj_forward"; - -(*Version for removal of duplicate literals*) -val major::prems = Goal - "[| P'|Q'; P' ==> P; [| Q'; P==>False |] ==> Q |] ==> P|Q"; -by (cut_facts_tac [major] 1); -by (blast_tac (claset() addIs prems) 1); -qed "disj_forward2"; - -val major::prems = Goal - "[| ALL x. P'(x); !!x. P'(x) ==> P(x) |] ==> ALL x. P(x)"; -by (rtac allI 1); -by (resolve_tac prems 1); -by (rtac (major RS spec) 1); -qed "all_forward"; - -val major::prems = Goal - "[| EX x. P'(x); !!x. P'(x) ==> P(x) |] ==> EX x. P(x)"; -by (rtac (major RS exE) 1); -by (rtac exI 1); -by (eresolve_tac prems 1); -qed "ex_forward";