src/HOL/Hilbert_Choice.thy
 author haftmann Mon Mar 01 13:40:23 2010 +0100 (2010-03-01) changeset 35416 d8d7d1b785af parent 35216 7641e8d831d2 child 39036 dff91b90d74c permissions -rw-r--r--
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
```     1 (*  Title:      HOL/Hilbert_Choice.thy
```
```     2     Author:     Lawrence C Paulson, Tobias Nipkow
```
```     3     Copyright   2001  University of Cambridge
```
```     4 *)
```
```     5
```
```     6 header {* Hilbert's Epsilon-Operator and the Axiom of Choice *}
```
```     7
```
```     8 theory Hilbert_Choice
```
```     9 imports Nat Wellfounded Plain
```
```    10 uses ("Tools/meson.ML") ("Tools/choice_specification.ML")
```
```    11 begin
```
```    12
```
```    13 subsection {* Hilbert's epsilon *}
```
```    14
```
```    15 axiomatization Eps :: "('a => bool) => 'a" where
```
```    16   someI: "P x ==> P (Eps P)"
```
```    17
```
```    18 syntax (epsilon)
```
```    19   "_Eps"        :: "[pttrn, bool] => 'a"    ("(3\<some>_./ _)" [0, 10] 10)
```
```    20 syntax (HOL)
```
```    21   "_Eps"        :: "[pttrn, bool] => 'a"    ("(3@ _./ _)" [0, 10] 10)
```
```    22 syntax
```
```    23   "_Eps"        :: "[pttrn, bool] => 'a"    ("(3SOME _./ _)" [0, 10] 10)
```
```    24 translations
```
```    25   "SOME x. P" == "CONST Eps (%x. P)"
```
```    26
```
```    27 print_translation {*
```
```    28   [(@{const_syntax Eps}, fn [Abs abs] =>
```
```    29       let val (x, t) = atomic_abs_tr' abs
```
```    30       in Syntax.const @{syntax_const "_Eps"} \$ x \$ t end)]
```
```    31 *} -- {* to avoid eta-contraction of body *}
```
```    32
```
```    33 definition inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where
```
```    34 "inv_into A f == %x. SOME y. y : A & f y = x"
```
```    35
```
```    36 abbreviation inv :: "('a => 'b) => ('b => 'a)" where
```
```    37 "inv == inv_into UNIV"
```
```    38
```
```    39
```
```    40 subsection {*Hilbert's Epsilon-operator*}
```
```    41
```
```    42 text{*Easier to apply than @{text someI} if the witness comes from an
```
```    43 existential formula*}
```
```    44 lemma someI_ex [elim?]: "\<exists>x. P x ==> P (SOME x. P x)"
```
```    45 apply (erule exE)
```
```    46 apply (erule someI)
```
```    47 done
```
```    48
```
```    49 text{*Easier to apply than @{text someI} because the conclusion has only one
```
```    50 occurrence of @{term P}.*}
```
```    51 lemma someI2: "[| P a;  !!x. P x ==> Q x |] ==> Q (SOME x. P x)"
```
```    52 by (blast intro: someI)
```
```    53
```
```    54 text{*Easier to apply than @{text someI2} if the witness comes from an
```
```    55 existential formula*}
```
```    56 lemma someI2_ex: "[| \<exists>a. P a; !!x. P x ==> Q x |] ==> Q (SOME x. P x)"
```
```    57 by (blast intro: someI2)
```
```    58
```
```    59 lemma some_equality [intro]:
```
```    60      "[| P a;  !!x. P x ==> x=a |] ==> (SOME x. P x) = a"
```
```    61 by (blast intro: someI2)
```
```    62
```
```    63 lemma some1_equality: "[| EX!x. P x; P a |] ==> (SOME x. P x) = a"
```
```    64 by blast
```
```    65
```
```    66 lemma some_eq_ex: "P (SOME x. P x) =  (\<exists>x. P x)"
```
```    67 by (blast intro: someI)
```
```    68
```
```    69 lemma some_eq_trivial [simp]: "(SOME y. y=x) = x"
```
```    70 apply (rule some_equality)
```
```    71 apply (rule refl, assumption)
```
```    72 done
```
```    73
```
```    74 lemma some_sym_eq_trivial [simp]: "(SOME y. x=y) = x"
```
```    75 apply (rule some_equality)
```
```    76 apply (rule refl)
```
```    77 apply (erule sym)
```
```    78 done
```
```    79
```
```    80
```
```    81 subsection{*Axiom of Choice, Proved Using the Description Operator*}
```
```    82
```
```    83 text{*Used in @{text "Tools/meson.ML"}*}
```
```    84 lemma choice: "\<forall>x. \<exists>y. Q x y ==> \<exists>f. \<forall>x. Q x (f x)"
```
```    85 by (fast elim: someI)
```
```    86
```
```    87 lemma bchoice: "\<forall>x\<in>S. \<exists>y. Q x y ==> \<exists>f. \<forall>x\<in>S. Q x (f x)"
```
```    88 by (fast elim: someI)
```
```    89
```
```    90
```
```    91 subsection {*Function Inverse*}
```
```    92
```
```    93 lemma inv_def: "inv f = (%y. SOME x. f x = y)"
```
```    94 by(simp add: inv_into_def)
```
```    95
```
```    96 lemma inv_into_into: "x : f ` A ==> inv_into A f x : A"
```
```    97 apply (simp add: inv_into_def)
```
```    98 apply (fast intro: someI2)
```
```    99 done
```
```   100
```
```   101 lemma inv_id [simp]: "inv id = id"
```
```   102 by (simp add: inv_into_def id_def)
```
```   103
```
```   104 lemma inv_into_f_f [simp]:
```
```   105   "[| inj_on f A;  x : A |] ==> inv_into A f (f x) = x"
```
```   106 apply (simp add: inv_into_def inj_on_def)
```
```   107 apply (blast intro: someI2)
```
```   108 done
```
```   109
```
```   110 lemma inv_f_f: "inj f ==> inv f (f x) = x"
```
```   111 by simp
```
```   112
```
```   113 lemma f_inv_into_f: "y : f`A  ==> f (inv_into A f y) = y"
```
```   114 apply (simp add: inv_into_def)
```
```   115 apply (fast intro: someI2)
```
```   116 done
```
```   117
```
```   118 lemma inv_into_f_eq: "[| inj_on f A; x : A; f x = y |] ==> inv_into A f y = x"
```
```   119 apply (erule subst)
```
```   120 apply (fast intro: inv_into_f_f)
```
```   121 done
```
```   122
```
```   123 lemma inv_f_eq: "[| inj f; f x = y |] ==> inv f y = x"
```
```   124 by (simp add:inv_into_f_eq)
```
```   125
```
```   126 lemma inj_imp_inv_eq: "[| inj f; ALL x. f(g x) = x |] ==> inv f = g"
```
```   127 by (blast intro: ext inv_into_f_eq)
```
```   128
```
```   129 text{*But is it useful?*}
```
```   130 lemma inj_transfer:
```
```   131   assumes injf: "inj f" and minor: "!!y. y \<in> range(f) ==> P(inv f y)"
```
```   132   shows "P x"
```
```   133 proof -
```
```   134   have "f x \<in> range f" by auto
```
```   135   hence "P(inv f (f x))" by (rule minor)
```
```   136   thus "P x" by (simp add: inv_into_f_f [OF injf])
```
```   137 qed
```
```   138
```
```   139 lemma inj_iff: "(inj f) = (inv f o f = id)"
```
```   140 apply (simp add: o_def expand_fun_eq)
```
```   141 apply (blast intro: inj_on_inverseI inv_into_f_f)
```
```   142 done
```
```   143
```
```   144 lemma inv_o_cancel[simp]: "inj f ==> inv f o f = id"
```
```   145 by (simp add: inj_iff)
```
```   146
```
```   147 lemma o_inv_o_cancel[simp]: "inj f ==> g o inv f o f = g"
```
```   148 by (simp add: o_assoc[symmetric])
```
```   149
```
```   150 lemma inv_into_image_cancel[simp]:
```
```   151   "inj_on f A ==> S <= A ==> inv_into A f ` f ` S = S"
```
```   152 by(fastsimp simp: image_def)
```
```   153
```
```   154 lemma inj_imp_surj_inv: "inj f ==> surj (inv f)"
```
```   155 by (blast intro: surjI inv_into_f_f)
```
```   156
```
```   157 lemma surj_f_inv_f: "surj f ==> f(inv f y) = y"
```
```   158 by (simp add: f_inv_into_f surj_range)
```
```   159
```
```   160 lemma inv_into_injective:
```
```   161   assumes eq: "inv_into A f x = inv_into A f y"
```
```   162       and x: "x: f`A"
```
```   163       and y: "y: f`A"
```
```   164   shows "x=y"
```
```   165 proof -
```
```   166   have "f (inv_into A f x) = f (inv_into A f y)" using eq by simp
```
```   167   thus ?thesis by (simp add: f_inv_into_f x y)
```
```   168 qed
```
```   169
```
```   170 lemma inj_on_inv_into: "B <= f`A ==> inj_on (inv_into A f) B"
```
```   171 by (blast intro: inj_onI dest: inv_into_injective injD)
```
```   172
```
```   173 lemma bij_betw_inv_into: "bij_betw f A B ==> bij_betw (inv_into A f) B A"
```
```   174 by (auto simp add: bij_betw_def inj_on_inv_into)
```
```   175
```
```   176 lemma surj_imp_inj_inv: "surj f ==> inj (inv f)"
```
```   177 by (simp add: inj_on_inv_into surj_range)
```
```   178
```
```   179 lemma surj_iff: "(surj f) = (f o inv f = id)"
```
```   180 apply (simp add: o_def expand_fun_eq)
```
```   181 apply (blast intro: surjI surj_f_inv_f)
```
```   182 done
```
```   183
```
```   184 lemma surj_imp_inv_eq: "[| surj f; \<forall>x. g(f x) = x |] ==> inv f = g"
```
```   185 apply (rule ext)
```
```   186 apply (drule_tac x = "inv f x" in spec)
```
```   187 apply (simp add: surj_f_inv_f)
```
```   188 done
```
```   189
```
```   190 lemma bij_imp_bij_inv: "bij f ==> bij (inv f)"
```
```   191 by (simp add: bij_def inj_imp_surj_inv surj_imp_inj_inv)
```
```   192
```
```   193 lemma inv_equality: "[| !!x. g (f x) = x;  !!y. f (g y) = y |] ==> inv f = g"
```
```   194 apply (rule ext)
```
```   195 apply (auto simp add: inv_into_def)
```
```   196 done
```
```   197
```
```   198 lemma inv_inv_eq: "bij f ==> inv (inv f) = f"
```
```   199 apply (rule inv_equality)
```
```   200 apply (auto simp add: bij_def surj_f_inv_f)
```
```   201 done
```
```   202
```
```   203 (** bij(inv f) implies little about f.  Consider f::bool=>bool such that
```
```   204     f(True)=f(False)=True.  Then it's consistent with axiom someI that
```
```   205     inv f could be any function at all, including the identity function.
```
```   206     If inv f=id then inv f is a bijection, but inj f, surj(f) and
```
```   207     inv(inv f)=f all fail.
```
```   208 **)
```
```   209
```
```   210 lemma inv_into_comp:
```
```   211   "[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==>
```
```   212   inv_into A (f o g) x = (inv_into A g o inv_into (g ` A) f) x"
```
```   213 apply (rule inv_into_f_eq)
```
```   214   apply (fast intro: comp_inj_on)
```
```   215  apply (simp add: inv_into_into)
```
```   216 apply (simp add: f_inv_into_f inv_into_into)
```
```   217 done
```
```   218
```
```   219 lemma o_inv_distrib: "[| bij f; bij g |] ==> inv (f o g) = inv g o inv f"
```
```   220 apply (rule inv_equality)
```
```   221 apply (auto simp add: bij_def surj_f_inv_f)
```
```   222 done
```
```   223
```
```   224 lemma image_surj_f_inv_f: "surj f ==> f ` (inv f ` A) = A"
```
```   225 by (simp add: image_eq_UN surj_f_inv_f)
```
```   226
```
```   227 lemma image_inv_f_f: "inj f ==> (inv f) ` (f ` A) = A"
```
```   228 by (simp add: image_eq_UN)
```
```   229
```
```   230 lemma inv_image_comp: "inj f ==> inv f ` (f`X) = X"
```
```   231 by (auto simp add: image_def)
```
```   232
```
```   233 lemma bij_image_Collect_eq: "bij f ==> f ` Collect P = {y. P (inv f y)}"
```
```   234 apply auto
```
```   235 apply (force simp add: bij_is_inj)
```
```   236 apply (blast intro: bij_is_surj [THEN surj_f_inv_f, symmetric])
```
```   237 done
```
```   238
```
```   239 lemma bij_vimage_eq_inv_image: "bij f ==> f -` A = inv f ` A"
```
```   240 apply (auto simp add: bij_is_surj [THEN surj_f_inv_f])
```
```   241 apply (blast intro: bij_is_inj [THEN inv_into_f_f, symmetric])
```
```   242 done
```
```   243
```
```   244 lemma finite_fun_UNIVD1:
```
```   245   assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
```
```   246   and card: "card (UNIV :: 'b set) \<noteq> Suc 0"
```
```   247   shows "finite (UNIV :: 'a set)"
```
```   248 proof -
```
```   249   from fin have finb: "finite (UNIV :: 'b set)" by (rule finite_fun_UNIVD2)
```
```   250   with card have "card (UNIV :: 'b set) \<ge> Suc (Suc 0)"
```
```   251     by (cases "card (UNIV :: 'b set)") (auto simp add: card_eq_0_iff)
```
```   252   then obtain n where "card (UNIV :: 'b set) = Suc (Suc n)" "n = card (UNIV :: 'b set) - Suc (Suc 0)" by auto
```
```   253   then obtain b1 b2 where b1b2: "(b1 :: 'b) \<noteq> (b2 :: 'b)" by (auto simp add: card_Suc_eq)
```
```   254   from fin have "finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1))" by (rule finite_imageI)
```
```   255   moreover have "UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1)"
```
```   256   proof (rule UNIV_eq_I)
```
```   257     fix x :: 'a
```
```   258     from b1b2 have "x = inv (\<lambda>y. if y = x then b1 else b2) b1" by (simp add: inv_into_def)
```
```   259     thus "x \<in> range (\<lambda>f\<Colon>'a \<Rightarrow> 'b. inv f b1)" by blast
```
```   260   qed
```
```   261   ultimately show "finite (UNIV :: 'a set)" by simp
```
```   262 qed
```
```   263
```
```   264
```
```   265 subsection {*Other Consequences of Hilbert's Epsilon*}
```
```   266
```
```   267 text {*Hilbert's Epsilon and the @{term split} Operator*}
```
```   268
```
```   269 text{*Looping simprule*}
```
```   270 lemma split_paired_Eps: "(SOME x. P x) = (SOME (a,b). P(a,b))"
```
```   271   by simp
```
```   272
```
```   273 lemma Eps_split: "Eps (split P) = (SOME xy. P (fst xy) (snd xy))"
```
```   274   by (simp add: split_def)
```
```   275
```
```   276 lemma Eps_split_eq [simp]: "(@(x',y'). x = x' & y = y') = (x,y)"
```
```   277   by blast
```
```   278
```
```   279
```
```   280 text{*A relation is wellfounded iff it has no infinite descending chain*}
```
```   281 lemma wf_iff_no_infinite_down_chain:
```
```   282   "wf r = (~(\<exists>f. \<forall>i. (f(Suc i),f i) \<in> r))"
```
```   283 apply (simp only: wf_eq_minimal)
```
```   284 apply (rule iffI)
```
```   285  apply (rule notI)
```
```   286  apply (erule exE)
```
```   287  apply (erule_tac x = "{w. \<exists>i. w=f i}" in allE, blast)
```
```   288 apply (erule contrapos_np, simp, clarify)
```
```   289 apply (subgoal_tac "\<forall>n. nat_rec x (%i y. @z. z:Q & (z,y) :r) n \<in> Q")
```
```   290  apply (rule_tac x = "nat_rec x (%i y. @z. z:Q & (z,y) :r)" in exI)
```
```   291  apply (rule allI, simp)
```
```   292  apply (rule someI2_ex, blast, blast)
```
```   293 apply (rule allI)
```
```   294 apply (induct_tac "n", simp_all)
```
```   295 apply (rule someI2_ex, blast+)
```
```   296 done
```
```   297
```
```   298 lemma wf_no_infinite_down_chainE:
```
```   299   assumes "wf r" obtains k where "(f (Suc k), f k) \<notin> r"
```
```   300 using `wf r` wf_iff_no_infinite_down_chain[of r] by blast
```
```   301
```
```   302
```
```   303 text{*A dynamically-scoped fact for TFL *}
```
```   304 lemma tfl_some: "\<forall>P x. P x --> P (Eps P)"
```
```   305   by (blast intro: someI)
```
```   306
```
```   307
```
```   308 subsection {* Least value operator *}
```
```   309
```
```   310 definition
```
```   311   LeastM :: "['a => 'b::ord, 'a => bool] => 'a" where
```
```   312   "LeastM m P == SOME x. P x & (\<forall>y. P y --> m x <= m y)"
```
```   313
```
```   314 syntax
```
```   315   "_LeastM" :: "[pttrn, 'a => 'b::ord, bool] => 'a"    ("LEAST _ WRT _. _" [0, 4, 10] 10)
```
```   316 translations
```
```   317   "LEAST x WRT m. P" == "CONST LeastM m (%x. P)"
```
```   318
```
```   319 lemma LeastMI2:
```
```   320   "P x ==> (!!y. P y ==> m x <= m y)
```
```   321     ==> (!!x. P x ==> \<forall>y. P y --> m x \<le> m y ==> Q x)
```
```   322     ==> Q (LeastM m P)"
```
```   323   apply (simp add: LeastM_def)
```
```   324   apply (rule someI2_ex, blast, blast)
```
```   325   done
```
```   326
```
```   327 lemma LeastM_equality:
```
```   328   "P k ==> (!!x. P x ==> m k <= m x)
```
```   329     ==> m (LEAST x WRT m. P x) = (m k::'a::order)"
```
```   330   apply (rule LeastMI2, assumption, blast)
```
```   331   apply (blast intro!: order_antisym)
```
```   332   done
```
```   333
```
```   334 lemma wf_linord_ex_has_least:
```
```   335   "wf r ==> \<forall>x y. ((x,y):r^+) = ((y,x)~:r^*) ==> P k
```
```   336     ==> \<exists>x. P x & (!y. P y --> (m x,m y):r^*)"
```
```   337   apply (drule wf_trancl [THEN wf_eq_minimal [THEN iffD1]])
```
```   338   apply (drule_tac x = "m`Collect P" in spec, force)
```
```   339   done
```
```   340
```
```   341 lemma ex_has_least_nat:
```
```   342     "P k ==> \<exists>x. P x & (\<forall>y. P y --> m x <= (m y::nat))"
```
```   343   apply (simp only: pred_nat_trancl_eq_le [symmetric])
```
```   344   apply (rule wf_pred_nat [THEN wf_linord_ex_has_least])
```
```   345    apply (simp add: less_eq linorder_not_le pred_nat_trancl_eq_le, assumption)
```
```   346   done
```
```   347
```
```   348 lemma LeastM_nat_lemma:
```
```   349     "P k ==> P (LeastM m P) & (\<forall>y. P y --> m (LeastM m P) <= (m y::nat))"
```
```   350   apply (simp add: LeastM_def)
```
```   351   apply (rule someI_ex)
```
```   352   apply (erule ex_has_least_nat)
```
```   353   done
```
```   354
```
```   355 lemmas LeastM_natI = LeastM_nat_lemma [THEN conjunct1, standard]
```
```   356
```
```   357 lemma LeastM_nat_le: "P x ==> m (LeastM m P) <= (m x::nat)"
```
```   358 by (rule LeastM_nat_lemma [THEN conjunct2, THEN spec, THEN mp], assumption, assumption)
```
```   359
```
```   360
```
```   361 subsection {* Greatest value operator *}
```
```   362
```
```   363 definition
```
```   364   GreatestM :: "['a => 'b::ord, 'a => bool] => 'a" where
```
```   365   "GreatestM m P == SOME x. P x & (\<forall>y. P y --> m y <= m x)"
```
```   366
```
```   367 definition
```
```   368   Greatest :: "('a::ord => bool) => 'a" (binder "GREATEST " 10) where
```
```   369   "Greatest == GreatestM (%x. x)"
```
```   370
```
```   371 syntax
```
```   372   "_GreatestM" :: "[pttrn, 'a => 'b::ord, bool] => 'a"
```
```   373       ("GREATEST _ WRT _. _" [0, 4, 10] 10)
```
```   374 translations
```
```   375   "GREATEST x WRT m. P" == "CONST GreatestM m (%x. P)"
```
```   376
```
```   377 lemma GreatestMI2:
```
```   378   "P x ==> (!!y. P y ==> m y <= m x)
```
```   379     ==> (!!x. P x ==> \<forall>y. P y --> m y \<le> m x ==> Q x)
```
```   380     ==> Q (GreatestM m P)"
```
```   381   apply (simp add: GreatestM_def)
```
```   382   apply (rule someI2_ex, blast, blast)
```
```   383   done
```
```   384
```
```   385 lemma GreatestM_equality:
```
```   386  "P k ==> (!!x. P x ==> m x <= m k)
```
```   387     ==> m (GREATEST x WRT m. P x) = (m k::'a::order)"
```
```   388   apply (rule_tac m = m in GreatestMI2, assumption, blast)
```
```   389   apply (blast intro!: order_antisym)
```
```   390   done
```
```   391
```
```   392 lemma Greatest_equality:
```
```   393   "P (k::'a::order) ==> (!!x. P x ==> x <= k) ==> (GREATEST x. P x) = k"
```
```   394   apply (simp add: Greatest_def)
```
```   395   apply (erule GreatestM_equality, blast)
```
```   396   done
```
```   397
```
```   398 lemma ex_has_greatest_nat_lemma:
```
```   399   "P k ==> \<forall>x. P x --> (\<exists>y. P y & ~ ((m y::nat) <= m x))
```
```   400     ==> \<exists>y. P y & ~ (m y < m k + n)"
```
```   401   apply (induct n, force)
```
```   402   apply (force simp add: le_Suc_eq)
```
```   403   done
```
```   404
```
```   405 lemma ex_has_greatest_nat:
```
```   406   "P k ==> \<forall>y. P y --> m y < b
```
```   407     ==> \<exists>x. P x & (\<forall>y. P y --> (m y::nat) <= m x)"
```
```   408   apply (rule ccontr)
```
```   409   apply (cut_tac P = P and n = "b - m k" in ex_has_greatest_nat_lemma)
```
```   410     apply (subgoal_tac  "m k <= b", auto)
```
```   411   done
```
```   412
```
```   413 lemma GreatestM_nat_lemma:
```
```   414   "P k ==> \<forall>y. P y --> m y < b
```
```   415     ==> P (GreatestM m P) & (\<forall>y. P y --> (m y::nat) <= m (GreatestM m P))"
```
```   416   apply (simp add: GreatestM_def)
```
```   417   apply (rule someI_ex)
```
```   418   apply (erule ex_has_greatest_nat, assumption)
```
```   419   done
```
```   420
```
```   421 lemmas GreatestM_natI = GreatestM_nat_lemma [THEN conjunct1, standard]
```
```   422
```
```   423 lemma GreatestM_nat_le:
```
```   424   "P x ==> \<forall>y. P y --> m y < b
```
```   425     ==> (m x::nat) <= m (GreatestM m P)"
```
```   426   apply (blast dest: GreatestM_nat_lemma [THEN conjunct2, THEN spec, of P])
```
```   427   done
```
```   428
```
```   429
```
```   430 text {* \medskip Specialization to @{text GREATEST}. *}
```
```   431
```
```   432 lemma GreatestI: "P (k::nat) ==> \<forall>y. P y --> y < b ==> P (GREATEST x. P x)"
```
```   433   apply (simp add: Greatest_def)
```
```   434   apply (rule GreatestM_natI, auto)
```
```   435   done
```
```   436
```
```   437 lemma Greatest_le:
```
```   438     "P x ==> \<forall>y. P y --> y < b ==> (x::nat) <= (GREATEST x. P x)"
```
```   439   apply (simp add: Greatest_def)
```
```   440   apply (rule GreatestM_nat_le, auto)
```
```   441   done
```
```   442
```
```   443
```
```   444 subsection {* The Meson proof procedure *}
```
```   445
```
```   446 subsubsection {* Negation Normal Form *}
```
```   447
```
```   448 text {* de Morgan laws *}
```
```   449
```
```   450 lemma meson_not_conjD: "~(P&Q) ==> ~P | ~Q"
```
```   451   and meson_not_disjD: "~(P|Q) ==> ~P & ~Q"
```
```   452   and meson_not_notD: "~~P ==> P"
```
```   453   and meson_not_allD: "!!P. ~(\<forall>x. P(x)) ==> \<exists>x. ~P(x)"
```
```   454   and meson_not_exD: "!!P. ~(\<exists>x. P(x)) ==> \<forall>x. ~P(x)"
```
```   455   by fast+
```
```   456
```
```   457 text {* Removal of @{text "-->"} and @{text "<->"} (positive and
```
```   458 negative occurrences) *}
```
```   459
```
```   460 lemma meson_imp_to_disjD: "P-->Q ==> ~P | Q"
```
```   461   and meson_not_impD: "~(P-->Q) ==> P & ~Q"
```
```   462   and meson_iff_to_disjD: "P=Q ==> (~P | Q) & (~Q | P)"
```
```   463   and meson_not_iffD: "~(P=Q) ==> (P | Q) & (~P | ~Q)"
```
```   464     -- {* Much more efficient than @{prop "(P & ~Q) | (Q & ~P)"} for computing CNF *}
```
```   465   and meson_not_refl_disj_D: "x ~= x | P ==> P"
```
```   466   by fast+
```
```   467
```
```   468
```
```   469 subsubsection {* Pulling out the existential quantifiers *}
```
```   470
```
```   471 text {* Conjunction *}
```
```   472
```
```   473 lemma meson_conj_exD1: "!!P Q. (\<exists>x. P(x)) & Q ==> \<exists>x. P(x) & Q"
```
```   474   and meson_conj_exD2: "!!P Q. P & (\<exists>x. Q(x)) ==> \<exists>x. P & Q(x)"
```
```   475   by fast+
```
```   476
```
```   477
```
```   478 text {* Disjunction *}
```
```   479
```
```   480 lemma meson_disj_exD: "!!P Q. (\<exists>x. P(x)) | (\<exists>x. Q(x)) ==> \<exists>x. P(x) | Q(x)"
```
```   481   -- {* DO NOT USE with forall-Skolemization: makes fewer schematic variables!! *}
```
```   482   -- {* With ex-Skolemization, makes fewer Skolem constants *}
```
```   483   and meson_disj_exD1: "!!P Q. (\<exists>x. P(x)) | Q ==> \<exists>x. P(x) | Q"
```
```   484   and meson_disj_exD2: "!!P Q. P | (\<exists>x. Q(x)) ==> \<exists>x. P | Q(x)"
```
```   485   by fast+
```
```   486
```
```   487
```
```   488 subsubsection {* Generating clauses for the Meson Proof Procedure *}
```
```   489
```
```   490 text {* Disjunctions *}
```
```   491
```
```   492 lemma meson_disj_assoc: "(P|Q)|R ==> P|(Q|R)"
```
```   493   and meson_disj_comm: "P|Q ==> Q|P"
```
```   494   and meson_disj_FalseD1: "False|P ==> P"
```
```   495   and meson_disj_FalseD2: "P|False ==> P"
```
```   496   by fast+
```
```   497
```
```   498
```
```   499 subsection{*Lemmas for Meson, the Model Elimination Procedure*}
```
```   500
```
```   501 text{* Generation of contrapositives *}
```
```   502
```
```   503 text{*Inserts negated disjunct after removing the negation; P is a literal.
```
```   504   Model elimination requires assuming the negation of every attempted subgoal,
```
```   505   hence the negated disjuncts.*}
```
```   506 lemma make_neg_rule: "~P|Q ==> ((~P==>P) ==> Q)"
```
```   507 by blast
```
```   508
```
```   509 text{*Version for Plaisted's "Postive refinement" of the Meson procedure*}
```
```   510 lemma make_refined_neg_rule: "~P|Q ==> (P ==> Q)"
```
```   511 by blast
```
```   512
```
```   513 text{*@{term P} should be a literal*}
```
```   514 lemma make_pos_rule: "P|Q ==> ((P==>~P) ==> Q)"
```
```   515 by blast
```
```   516
```
```   517 text{*Versions of @{text make_neg_rule} and @{text make_pos_rule} that don't
```
```   518 insert new assumptions, for ordinary resolution.*}
```
```   519
```
```   520 lemmas make_neg_rule' = make_refined_neg_rule
```
```   521
```
```   522 lemma make_pos_rule': "[|P|Q; ~P|] ==> Q"
```
```   523 by blast
```
```   524
```
```   525 text{* Generation of a goal clause -- put away the final literal *}
```
```   526
```
```   527 lemma make_neg_goal: "~P ==> ((~P==>P) ==> False)"
```
```   528 by blast
```
```   529
```
```   530 lemma make_pos_goal: "P ==> ((P==>~P) ==> False)"
```
```   531 by blast
```
```   532
```
```   533
```
```   534 subsubsection{* Lemmas for Forward Proof*}
```
```   535
```
```   536 text{*There is a similarity to congruence rules*}
```
```   537
```
```   538 (*NOTE: could handle conjunctions (faster?) by
```
```   539     nf(th RS conjunct2) RS (nf(th RS conjunct1) RS conjI) *)
```
```   540 lemma conj_forward: "[| P'&Q';  P' ==> P;  Q' ==> Q |] ==> P&Q"
```
```   541 by blast
```
```   542
```
```   543 lemma disj_forward: "[| P'|Q';  P' ==> P;  Q' ==> Q |] ==> P|Q"
```
```   544 by blast
```
```   545
```
```   546 (*Version of @{text disj_forward} for removal of duplicate literals*)
```
```   547 lemma disj_forward2:
```
```   548     "[| P'|Q';  P' ==> P;  [| Q'; P==>False |] ==> Q |] ==> P|Q"
```
```   549 apply blast
```
```   550 done
```
```   551
```
```   552 lemma all_forward: "[| \<forall>x. P'(x);  !!x. P'(x) ==> P(x) |] ==> \<forall>x. P(x)"
```
```   553 by blast
```
```   554
```
```   555 lemma ex_forward: "[| \<exists>x. P'(x);  !!x. P'(x) ==> P(x) |] ==> \<exists>x. P(x)"
```
```   556 by blast
```
```   557
```
```   558
```
```   559 subsection {* Meson package *}
```
```   560
```
```   561 use "Tools/meson.ML"
```
```   562
```
```   563 setup Meson.setup
```
```   564
```
```   565
```
```   566 subsection {* Specification package -- Hilbertized version *}
```
```   567
```
```   568 lemma exE_some: "[| Ex P ; c == Eps P |] ==> P c"
```
```   569   by (simp only: someI_ex)
```
```   570
```
```   571 use "Tools/choice_specification.ML"
```
```   572
```
```   573
```
```   574 end
```