src/HOL/Hilbert_Choice.thy
changeset 14760 a08e916f4946
parent 14399 dc677b35e54f
child 14872 3f2144aebd76
     1.1 --- a/src/HOL/Hilbert_Choice.thy	Wed May 19 11:24:54 2004 +0200
     1.2 +++ b/src/HOL/Hilbert_Choice.thy	Wed May 19 11:29:47 2004 +0200
     1.3 @@ -1,13 +1,13 @@
     1.4  (*  Title:      HOL/Hilbert_Choice.thy
     1.5 -    ID:         $Id$
     1.6 +    ID: $Id$
     1.7      Author:     Lawrence C Paulson
     1.8      Copyright   2001  University of Cambridge
     1.9  *)
    1.10  
    1.11 -header {* Hilbert's epsilon-operator and everything to do with the Axiom of Choice *}
    1.12 +header {* Hilbert's Epsilon-Operator and the Axiom of Choice *}
    1.13  
    1.14  theory Hilbert_Choice = NatArith
    1.15 -files ("Hilbert_Choice_lemmas.ML") ("meson_lemmas.ML") ("Tools/meson.ML") ("Tools/specification_package.ML"):
    1.16 +files ("Tools/meson.ML") ("Tools/specification_package.ML"):
    1.17  
    1.18  
    1.19  subsection {* Hilbert's epsilon *}
    1.20 @@ -40,26 +40,217 @@
    1.21    "inv(f :: 'a => 'b) == %y. SOME x. f x = y"
    1.22  
    1.23    Inv :: "'a set => ('a => 'b) => ('b => 'a)"
    1.24 -  "Inv A f == %x. SOME y. y : A & f y = x"
    1.25 +  "Inv A f == %x. SOME y. y \<in> A & f y = x"
    1.26 +
    1.27 +
    1.28 +subsection {*Hilbert's Epsilon-operator*}
    1.29 +
    1.30 +text{*Easier to apply than @{text someI} if the witness comes from an
    1.31 +existential formula*}
    1.32 +lemma someI_ex [elim?]: "\<exists>x. P x ==> P (SOME x. P x)"
    1.33 +apply (erule exE)
    1.34 +apply (erule someI)
    1.35 +done
    1.36 +
    1.37 +text{*Easier to apply than @{text someI} because the conclusion has only one
    1.38 +occurrence of @{term P}.*}
    1.39 +lemma someI2: "[| P a;  !!x. P x ==> Q x |] ==> Q (SOME x. P x)"
    1.40 +by (blast intro: someI)
    1.41 +
    1.42 +text{*Easier to apply than @{text someI2} if the witness comes from an
    1.43 +existential formula*}
    1.44 +lemma someI2_ex: "[| \<exists>a. P a; !!x. P x ==> Q x |] ==> Q (SOME x. P x)"
    1.45 +by (blast intro: someI2)
    1.46 +
    1.47 +lemma some_equality [intro]:
    1.48 +     "[| P a;  !!x. P x ==> x=a |] ==> (SOME x. P x) = a"
    1.49 +by (blast intro: someI2)
    1.50 +
    1.51 +lemma some1_equality: "[| EX!x. P x; P a |] ==> (SOME x. P x) = a"
    1.52 +by (blast intro: some_equality)
    1.53 +
    1.54 +lemma some_eq_ex: "P (SOME x. P x) =  (\<exists>x. P x)"
    1.55 +by (blast intro: someI)
    1.56 +
    1.57 +lemma some_eq_trivial [simp]: "(SOME y. y=x) = x"
    1.58 +apply (rule some_equality)
    1.59 +apply (rule refl, assumption)
    1.60 +done
    1.61 +
    1.62 +lemma some_sym_eq_trivial [simp]: "(SOME y. x=y) = x"
    1.63 +apply (rule some_equality)
    1.64 +apply (rule refl)
    1.65 +apply (erule sym)
    1.66 +done
    1.67 +
    1.68 +
    1.69 +subsection{*Axiom of Choice, Proved Using the Description Operator*}
    1.70 +
    1.71 +text{*Used in @{text "Tools/meson.ML"}*}
    1.72 +lemma choice: "\<forall>x. \<exists>y. Q x y ==> \<exists>f. \<forall>x. Q x (f x)"
    1.73 +by (fast elim: someI)
    1.74 +
    1.75 +lemma bchoice: "\<forall>x\<in>S. \<exists>y. Q x y ==> \<exists>f. \<forall>x\<in>S. Q x (f x)"
    1.76 +by (fast elim: someI)
    1.77 +
    1.78 +
    1.79 +subsection {*Function Inverse*}
    1.80 +
    1.81 +lemma inv_id [simp]: "inv id = id"
    1.82 +by (simp add: inv_def id_def)
    1.83 +
    1.84 +text{*A one-to-one function has an inverse.*}
    1.85 +lemma inv_f_f [simp]: "inj f ==> inv f (f x) = x"
    1.86 +by (simp add: inv_def inj_eq)
    1.87 +
    1.88 +lemma inv_f_eq: "[| inj f;  f x = y |] ==> inv f y = x"
    1.89 +apply (erule subst)
    1.90 +apply (erule inv_f_f)
    1.91 +done
    1.92 +
    1.93 +lemma inj_imp_inv_eq: "[| inj f; \<forall>x. f(g x) = x |] ==> inv f = g"
    1.94 +by (blast intro: ext inv_f_eq)
    1.95 +
    1.96 +text{*But is it useful?*}
    1.97 +lemma inj_transfer:
    1.98 +  assumes injf: "inj f" and minor: "!!y. y \<in> range(f) ==> P(inv f y)"
    1.99 +  shows "P x"
   1.100 +proof -
   1.101 +  have "f x \<in> range f" by auto
   1.102 +  hence "P(inv f (f x))" by (rule minor)
   1.103 +  thus "P x" by (simp add: inv_f_f [OF injf])
   1.104 +qed
   1.105  
   1.106  
   1.107 -use "Hilbert_Choice_lemmas.ML"
   1.108 -declare someI_ex [elim?];
   1.109 +lemma inj_iff: "(inj f) = (inv f o f = id)"
   1.110 +apply (simp add: o_def expand_fun_eq)
   1.111 +apply (blast intro: inj_on_inverseI inv_f_f)
   1.112 +done
   1.113 +
   1.114 +lemma inj_imp_surj_inv: "inj f ==> surj (inv f)"
   1.115 +by (blast intro: surjI inv_f_f)
   1.116 +
   1.117 +lemma f_inv_f: "y \<in> range(f) ==> f(inv f y) = y"
   1.118 +apply (simp add: inv_def)
   1.119 +apply (fast intro: someI)
   1.120 +done
   1.121 +
   1.122 +lemma surj_f_inv_f: "surj f ==> f(inv f y) = y"
   1.123 +by (simp add: f_inv_f surj_range)
   1.124 +
   1.125 +lemma inv_injective:
   1.126 +  assumes eq: "inv f x = inv f y"
   1.127 +      and x: "x: range f"
   1.128 +      and y: "y: range f"
   1.129 +  shows "x=y"
   1.130 +proof -
   1.131 +  have "f (inv f x) = f (inv f y)" using eq by simp
   1.132 +  thus ?thesis by (simp add: f_inv_f x y) 
   1.133 +qed
   1.134 +
   1.135 +lemma inj_on_inv: "A <= range(f) ==> inj_on (inv f) A"
   1.136 +by (fast intro: inj_onI elim: inv_injective injD)
   1.137 +
   1.138 +lemma surj_imp_inj_inv: "surj f ==> inj (inv f)"
   1.139 +by (simp add: inj_on_inv surj_range)
   1.140 +
   1.141 +lemma surj_iff: "(surj f) = (f o inv f = id)"
   1.142 +apply (simp add: o_def expand_fun_eq)
   1.143 +apply (blast intro: surjI surj_f_inv_f)
   1.144 +done
   1.145 +
   1.146 +lemma surj_imp_inv_eq: "[| surj f; \<forall>x. g(f x) = x |] ==> inv f = g"
   1.147 +apply (rule ext)
   1.148 +apply (drule_tac x = "inv f x" in spec)
   1.149 +apply (simp add: surj_f_inv_f)
   1.150 +done
   1.151 +
   1.152 +lemma bij_imp_bij_inv: "bij f ==> bij (inv f)"
   1.153 +by (simp add: bij_def inj_imp_surj_inv surj_imp_inj_inv)
   1.154  
   1.155 -lemma Inv_mem: "[| f ` A = B;  x \<in> B |] ==> Inv A f x \<in> A"
   1.156 -apply (unfold Inv_def)
   1.157 +lemma inv_equality: "[| !!x. g (f x) = x;  !!y. f (g y) = y |] ==> inv f = g"
   1.158 +apply (rule ext)
   1.159 +apply (auto simp add: inv_def)
   1.160 +done
   1.161 +
   1.162 +lemma inv_inv_eq: "bij f ==> inv (inv f) = f"
   1.163 +apply (rule inv_equality)
   1.164 +apply (auto simp add: bij_def surj_f_inv_f)
   1.165 +done
   1.166 +
   1.167 +(** bij(inv f) implies little about f.  Consider f::bool=>bool such that
   1.168 +    f(True)=f(False)=True.  Then it's consistent with axiom someI that
   1.169 +    inv f could be any function at all, including the identity function.
   1.170 +    If inv f=id then inv f is a bijection, but inj f, surj(f) and
   1.171 +    inv(inv f)=f all fail.
   1.172 +**)
   1.173 +
   1.174 +lemma o_inv_distrib: "[| bij f; bij g |] ==> inv (f o g) = inv g o inv f"
   1.175 +apply (rule inv_equality)
   1.176 +apply (auto simp add: bij_def surj_f_inv_f)
   1.177 +done
   1.178 +
   1.179 +
   1.180 +lemma image_surj_f_inv_f: "surj f ==> f ` (inv f ` A) = A"
   1.181 +by (simp add: image_eq_UN surj_f_inv_f)
   1.182 +
   1.183 +lemma image_inv_f_f: "inj f ==> (inv f) ` (f ` A) = A"
   1.184 +by (simp add: image_eq_UN)
   1.185 +
   1.186 +lemma inv_image_comp: "inj f ==> inv f ` (f`X) = X"
   1.187 +by (auto simp add: image_def)
   1.188 +
   1.189 +lemma bij_image_Collect_eq: "bij f ==> f ` Collect P = {y. P (inv f y)}"
   1.190 +apply auto
   1.191 +apply (force simp add: bij_is_inj)
   1.192 +apply (blast intro: bij_is_surj [THEN surj_f_inv_f, symmetric])
   1.193 +done
   1.194 +
   1.195 +lemma bij_vimage_eq_inv_image: "bij f ==> f -` A = inv f ` A" 
   1.196 +apply (auto simp add: bij_is_surj [THEN surj_f_inv_f])
   1.197 +apply (blast intro: bij_is_inj [THEN inv_f_f, symmetric])
   1.198 +done
   1.199 +
   1.200 +
   1.201 +subsection {*Inverse of a PI-function (restricted domain)*}
   1.202 +
   1.203 +lemma Inv_f_f: "[| inj_on f A;  x \<in> A |] ==> Inv A f (f x) = x"
   1.204 +apply (simp add: Inv_def inj_on_def)
   1.205 +apply (blast intro: someI2)
   1.206 +done
   1.207 +
   1.208 +lemma f_Inv_f: "y \<in> f`A  ==> f (Inv A f y) = y"
   1.209 +apply (simp add: Inv_def)
   1.210  apply (fast intro: someI2)
   1.211  done
   1.212  
   1.213 -lemma Inv_f_eq:
   1.214 -  "[| inj_on f A; f x = y; x : A |] ==> Inv A f y = x"
   1.215 +lemma Inv_injective:
   1.216 +  assumes eq: "Inv A f x = Inv A f y"
   1.217 +      and x: "x: f`A"
   1.218 +      and y: "y: f`A"
   1.219 +  shows "x=y"
   1.220 +proof -
   1.221 +  have "f (Inv A f x) = f (Inv A f y)" using eq by simp
   1.222 +  thus ?thesis by (simp add: f_Inv_f x y) 
   1.223 +qed
   1.224 +
   1.225 +lemma inj_on_Inv: "B <= f`A ==> inj_on (Inv A f) B"
   1.226 +apply (rule inj_onI)
   1.227 +apply (blast intro: inj_onI dest: Inv_injective injD)
   1.228 +done
   1.229 +
   1.230 +lemma Inv_mem: "[| f ` A = B;  x \<in> B |] ==> Inv A f x \<in> A"
   1.231 +apply (simp add: Inv_def)
   1.232 +apply (fast intro: someI2)
   1.233 +done
   1.234 +
   1.235 +lemma Inv_f_eq: "[| inj_on f A; f x = y; x \<in> A |] ==> Inv A f y = x"
   1.236    apply (erule subst)
   1.237 -  apply (erule Inv_f_f)
   1.238 -  apply assumption
   1.239 +  apply (erule Inv_f_f, assumption)
   1.240    done
   1.241  
   1.242  lemma Inv_comp:
   1.243 -  "[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==>
   1.244 +  "[| inj_on f (g ` A); inj_on g A; x \<in> f ` g ` A |] ==>
   1.245    Inv A (f o g) x = (Inv A g o Inv (g ` A) f) x"
   1.246    apply simp
   1.247    apply (rule Inv_f_eq)
   1.248 @@ -68,8 +259,42 @@
   1.249    apply (simp add: Inv_mem)
   1.250    done
   1.251  
   1.252 +
   1.253 +subsection {*Other Consequences of Hilbert's Epsilon*}
   1.254 +
   1.255 +text {*Hilbert's Epsilon and the @{term split} Operator*}
   1.256 +
   1.257 +text{*Looping simprule*}
   1.258 +lemma split_paired_Eps: "(SOME x. P x) = (SOME (a,b). P(a,b))"
   1.259 +by (simp add: split_Pair_apply)
   1.260 +
   1.261 +lemma Eps_split: "Eps (split P) = (SOME xy. P (fst xy) (snd xy))"
   1.262 +by (simp add: split_def)
   1.263 +
   1.264 +lemma Eps_split_eq [simp]: "(@(x',y'). x = x' & y = y') = (x,y)"
   1.265 +by blast
   1.266 +
   1.267 +
   1.268 +text{*A relation is wellfounded iff it has no infinite descending chain*}
   1.269 +lemma wf_iff_no_infinite_down_chain:
   1.270 +  "wf r = (~(\<exists>f. \<forall>i. (f(Suc i),f i) \<in> r))"
   1.271 +apply (simp only: wf_eq_minimal)
   1.272 +apply (rule iffI)
   1.273 + apply (rule notI)
   1.274 + apply (erule exE)
   1.275 + apply (erule_tac x = "{w. \<exists>i. w=f i}" in allE, blast)
   1.276 +apply (erule contrapos_np, simp, clarify)
   1.277 +apply (subgoal_tac "\<forall>n. nat_rec x (%i y. @z. z:Q & (z,y) :r) n \<in> Q")
   1.278 + apply (rule_tac x = "nat_rec x (%i y. @z. z:Q & (z,y) :r)" in exI)
   1.279 + apply (rule allI, simp)
   1.280 + apply (rule someI2_ex, blast, blast)
   1.281 +apply (rule allI)
   1.282 +apply (induct_tac "n", simp_all)
   1.283 +apply (rule someI2_ex, blast+)
   1.284 +done
   1.285 +
   1.286 +text{*A dynamically-scoped fact for TFL *}
   1.287  lemma tfl_some: "\<forall>P x. P x --> P (Eps P)"
   1.288 -  -- {* dynamically-scoped fact for TFL *}
   1.289    by (blast intro: someI)
   1.290  
   1.291  
   1.292 @@ -77,7 +302,7 @@
   1.293  
   1.294  constdefs
   1.295    LeastM :: "['a => 'b::ord, 'a => bool] => 'a"
   1.296 -  "LeastM m P == SOME x. P x & (ALL y. P y --> m x <= m y)"
   1.297 +  "LeastM m P == SOME x. P x & (\<forall>y. P y --> m x <= m y)"
   1.298  
   1.299  syntax
   1.300    "_LeastM" :: "[pttrn, 'a => 'b::ord, bool] => 'a"    ("LEAST _ WRT _. _" [0, 4, 10] 10)
   1.301 @@ -88,7 +313,7 @@
   1.302    "P x ==> (!!y. P y ==> m x <= m y)
   1.303      ==> (!!x. P x ==> \<forall>y. P y --> m x \<le> m y ==> Q x)
   1.304      ==> Q (LeastM m P)"
   1.305 -  apply (unfold LeastM_def)
   1.306 +  apply (simp add: LeastM_def)
   1.307    apply (rule someI2_ex, blast, blast)
   1.308    done
   1.309  
   1.310 @@ -100,22 +325,22 @@
   1.311    done
   1.312  
   1.313  lemma wf_linord_ex_has_least:
   1.314 -  "wf r ==> ALL x y. ((x,y):r^+) = ((y,x)~:r^*) ==> P k
   1.315 -    ==> EX x. P x & (!y. P y --> (m x,m y):r^*)"
   1.316 +  "wf r ==> \<forall>x y. ((x,y):r^+) = ((y,x)~:r^*) ==> P k
   1.317 +    ==> \<exists>x. P x & (!y. P y --> (m x,m y):r^*)"
   1.318    apply (drule wf_trancl [THEN wf_eq_minimal [THEN iffD1]])
   1.319    apply (drule_tac x = "m`Collect P" in spec, force)
   1.320    done
   1.321  
   1.322  lemma ex_has_least_nat:
   1.323 -    "P k ==> EX x. P x & (ALL y. P y --> m x <= (m y::nat))"
   1.324 +    "P k ==> \<exists>x. P x & (\<forall>y. P y --> m x <= (m y::nat))"
   1.325    apply (simp only: pred_nat_trancl_eq_le [symmetric])
   1.326    apply (rule wf_pred_nat [THEN wf_linord_ex_has_least])
   1.327     apply (simp add: less_eq not_le_iff_less pred_nat_trancl_eq_le, assumption)
   1.328    done
   1.329  
   1.330  lemma LeastM_nat_lemma:
   1.331 -    "P k ==> P (LeastM m P) & (ALL y. P y --> m (LeastM m P) <= (m y::nat))"
   1.332 -  apply (unfold LeastM_def)
   1.333 +    "P k ==> P (LeastM m P) & (\<forall>y. P y --> m (LeastM m P) <= (m y::nat))"
   1.334 +  apply (simp add: LeastM_def)
   1.335    apply (rule someI_ex)
   1.336    apply (erule ex_has_least_nat)
   1.337    done
   1.338 @@ -130,7 +355,7 @@
   1.339  
   1.340  constdefs
   1.341    GreatestM :: "['a => 'b::ord, 'a => bool] => 'a"
   1.342 -  "GreatestM m P == SOME x. P x & (ALL y. P y --> m y <= m x)"
   1.343 +  "GreatestM m P == SOME x. P x & (\<forall>y. P y --> m y <= m x)"
   1.344  
   1.345    Greatest :: "('a::ord => bool) => 'a"    (binder "GREATEST " 10)
   1.346    "Greatest == GreatestM (%x. x)"
   1.347 @@ -146,7 +371,7 @@
   1.348    "P x ==> (!!y. P y ==> m y <= m x)
   1.349      ==> (!!x. P x ==> \<forall>y. P y --> m y \<le> m x ==> Q x)
   1.350      ==> Q (GreatestM m P)"
   1.351 -  apply (unfold GreatestM_def)
   1.352 +  apply (simp add: GreatestM_def)
   1.353    apply (rule someI2_ex, blast, blast)
   1.354    done
   1.355  
   1.356 @@ -159,29 +384,29 @@
   1.357  
   1.358  lemma Greatest_equality:
   1.359    "P (k::'a::order) ==> (!!x. P x ==> x <= k) ==> (GREATEST x. P x) = k"
   1.360 -  apply (unfold Greatest_def)
   1.361 +  apply (simp add: Greatest_def)
   1.362    apply (erule GreatestM_equality, blast)
   1.363    done
   1.364  
   1.365  lemma ex_has_greatest_nat_lemma:
   1.366 -  "P k ==> ALL x. P x --> (EX y. P y & ~ ((m y::nat) <= m x))
   1.367 -    ==> EX y. P y & ~ (m y < m k + n)"
   1.368 +  "P k ==> \<forall>x. P x --> (\<exists>y. P y & ~ ((m y::nat) <= m x))
   1.369 +    ==> \<exists>y. P y & ~ (m y < m k + n)"
   1.370    apply (induct_tac n, force)
   1.371    apply (force simp add: le_Suc_eq)
   1.372    done
   1.373  
   1.374  lemma ex_has_greatest_nat:
   1.375 -  "P k ==> ALL y. P y --> m y < b
   1.376 -    ==> EX x. P x & (ALL y. P y --> (m y::nat) <= m x)"
   1.377 +  "P k ==> \<forall>y. P y --> m y < b
   1.378 +    ==> \<exists>x. P x & (\<forall>y. P y --> (m y::nat) <= m x)"
   1.379    apply (rule ccontr)
   1.380    apply (cut_tac P = P and n = "b - m k" in ex_has_greatest_nat_lemma)
   1.381      apply (subgoal_tac [3] "m k <= b", auto)
   1.382    done
   1.383  
   1.384  lemma GreatestM_nat_lemma:
   1.385 -  "P k ==> ALL y. P y --> m y < b
   1.386 -    ==> P (GreatestM m P) & (ALL y. P y --> (m y::nat) <= m (GreatestM m P))"
   1.387 -  apply (unfold GreatestM_def)
   1.388 +  "P k ==> \<forall>y. P y --> m y < b
   1.389 +    ==> P (GreatestM m P) & (\<forall>y. P y --> (m y::nat) <= m (GreatestM m P))"
   1.390 +  apply (simp add: GreatestM_def)
   1.391    apply (rule someI_ex)
   1.392    apply (erule ex_has_greatest_nat, assumption)
   1.393    done
   1.394 @@ -189,7 +414,7 @@
   1.395  lemmas GreatestM_natI = GreatestM_nat_lemma [THEN conjunct1, standard]
   1.396  
   1.397  lemma GreatestM_nat_le:
   1.398 -  "P x ==> ALL y. P y --> m y < b
   1.399 +  "P x ==> \<forall>y. P y --> m y < b
   1.400      ==> (m x::nat) <= m (GreatestM m P)"
   1.401    apply (blast dest: GreatestM_nat_lemma [THEN conjunct2, THEN spec])
   1.402    done
   1.403 @@ -197,14 +422,14 @@
   1.404  
   1.405  text {* \medskip Specialization to @{text GREATEST}. *}
   1.406  
   1.407 -lemma GreatestI: "P (k::nat) ==> ALL y. P y --> y < b ==> P (GREATEST x. P x)"
   1.408 -  apply (unfold Greatest_def)
   1.409 +lemma GreatestI: "P (k::nat) ==> \<forall>y. P y --> y < b ==> P (GREATEST x. P x)"
   1.410 +  apply (simp add: Greatest_def)
   1.411    apply (rule GreatestM_natI, auto)
   1.412    done
   1.413  
   1.414  lemma Greatest_le:
   1.415 -    "P x ==> ALL y. P y --> y < b ==> (x::nat) <= (GREATEST x. P x)"
   1.416 -  apply (unfold Greatest_def)
   1.417 +    "P x ==> \<forall>y. P y --> y < b ==> (x::nat) <= (GREATEST x. P x)"
   1.418 +  apply (simp add: Greatest_def)
   1.419    apply (rule GreatestM_nat_le, auto)
   1.420    done
   1.421  
   1.422 @@ -218,8 +443,8 @@
   1.423  lemma meson_not_conjD: "~(P&Q) ==> ~P | ~Q"
   1.424    and meson_not_disjD: "~(P|Q) ==> ~P & ~Q"
   1.425    and meson_not_notD: "~~P ==> P"
   1.426 -  and meson_not_allD: "!!P. ~(ALL x. P(x)) ==> EX x. ~P(x)"
   1.427 -  and meson_not_exD: "!!P. ~(EX x. P(x)) ==> ALL x. ~P(x)"
   1.428 +  and meson_not_allD: "!!P. ~(\<forall>x. P(x)) ==> \<exists>x. ~P(x)"
   1.429 +  and meson_not_exD: "!!P. ~(\<exists>x. P(x)) ==> \<forall>x. ~P(x)"
   1.430    by fast+
   1.431  
   1.432  text {* Removal of @{text "-->"} and @{text "<->"} (positive and
   1.433 @@ -237,18 +462,18 @@
   1.434  
   1.435  text {* Conjunction *}
   1.436  
   1.437 -lemma meson_conj_exD1: "!!P Q. (EX x. P(x)) & Q ==> EX x. P(x) & Q"
   1.438 -  and meson_conj_exD2: "!!P Q. P & (EX x. Q(x)) ==> EX x. P & Q(x)"
   1.439 +lemma meson_conj_exD1: "!!P Q. (\<exists>x. P(x)) & Q ==> \<exists>x. P(x) & Q"
   1.440 +  and meson_conj_exD2: "!!P Q. P & (\<exists>x. Q(x)) ==> \<exists>x. P & Q(x)"
   1.441    by fast+
   1.442  
   1.443  
   1.444  text {* Disjunction *}
   1.445  
   1.446 -lemma meson_disj_exD: "!!P Q. (EX x. P(x)) | (EX x. Q(x)) ==> EX x. P(x) | Q(x)"
   1.447 +lemma meson_disj_exD: "!!P Q. (\<exists>x. P(x)) | (\<exists>x. Q(x)) ==> \<exists>x. P(x) | Q(x)"
   1.448    -- {* DO NOT USE with forall-Skolemization: makes fewer schematic variables!! *}
   1.449    -- {* With ex-Skolemization, makes fewer Skolem constants *}
   1.450 -  and meson_disj_exD1: "!!P Q. (EX x. P(x)) | Q ==> EX x. P(x) | Q"
   1.451 -  and meson_disj_exD2: "!!P Q. P | (EX x. Q(x)) ==> EX x. P | Q(x)"
   1.452 +  and meson_disj_exD1: "!!P Q. (\<exists>x. P(x)) | Q ==> \<exists>x. P(x) | Q"
   1.453 +  and meson_disj_exD2: "!!P Q. P | (\<exists>x. Q(x)) ==> \<exists>x. P | Q(x)"
   1.454    by fast+
   1.455  
   1.456  
   1.457 @@ -262,7 +487,133 @@
   1.458    and meson_disj_FalseD2: "P|False ==> P"
   1.459    by fast+
   1.460  
   1.461 -use "meson_lemmas.ML"
   1.462 +
   1.463 +subsection{*Lemmas for Meson, the Model Elimination Procedure*}
   1.464 +
   1.465 +
   1.466 +text{* Generation of contrapositives *}
   1.467 +
   1.468 +text{*Inserts negated disjunct after removing the negation; P is a literal.
   1.469 +  Model elimination requires assuming the negation of every attempted subgoal,
   1.470 +  hence the negated disjuncts.*}
   1.471 +lemma make_neg_rule: "~P|Q ==> ((~P==>P) ==> Q)"
   1.472 +by blast
   1.473 +
   1.474 +text{*Version for Plaisted's "Postive refinement" of the Meson procedure*}
   1.475 +lemma make_refined_neg_rule: "~P|Q ==> (P ==> Q)"
   1.476 +by blast
   1.477 +
   1.478 +text{*@{term P} should be a literal*}
   1.479 +lemma make_pos_rule: "P|Q ==> ((P==>~P) ==> Q)"
   1.480 +by blast
   1.481 +
   1.482 +text{*Versions of @{text make_neg_rule} and @{text make_pos_rule} that don't
   1.483 +insert new assumptions, for ordinary resolution.*}
   1.484 +
   1.485 +lemmas make_neg_rule' = make_refined_neg_rule
   1.486 +
   1.487 +lemma make_pos_rule': "[|P|Q; ~P|] ==> Q"
   1.488 +by blast
   1.489 +
   1.490 +text{* Generation of a goal clause -- put away the final literal *}
   1.491 +
   1.492 +lemma make_neg_goal: "~P ==> ((~P==>P) ==> False)"
   1.493 +by blast
   1.494 +
   1.495 +lemma make_pos_goal: "P ==> ((P==>~P) ==> False)"
   1.496 +by blast
   1.497 +
   1.498 +
   1.499 +subsubsection{* Lemmas for Forward Proof*}
   1.500 +
   1.501 +text{*There is a similarity to congruence rules*}
   1.502 +
   1.503 +(*NOTE: could handle conjunctions (faster?) by
   1.504 +    nf(th RS conjunct2) RS (nf(th RS conjunct1) RS conjI) *)
   1.505 +lemma conj_forward: "[| P'&Q';  P' ==> P;  Q' ==> Q |] ==> P&Q"
   1.506 +by blast
   1.507 +
   1.508 +lemma disj_forward: "[| P'|Q';  P' ==> P;  Q' ==> Q |] ==> P|Q"
   1.509 +by blast
   1.510 +
   1.511 +(*Version of @{text disj_forward} for removal of duplicate literals*)
   1.512 +lemma disj_forward2:
   1.513 +    "[| P'|Q';  P' ==> P;  [| Q'; P==>False |] ==> Q |] ==> P|Q"
   1.514 +apply blast 
   1.515 +done
   1.516 +
   1.517 +lemma all_forward: "[| \<forall>x. P'(x);  !!x. P'(x) ==> P(x) |] ==> \<forall>x. P(x)"
   1.518 +by blast
   1.519 +
   1.520 +lemma ex_forward: "[| \<exists>x. P'(x);  !!x. P'(x) ==> P(x) |] ==> \<exists>x. P(x)"
   1.521 +by blast
   1.522 +
   1.523 +ML
   1.524 +{*
   1.525 +val inv_def = thm "inv_def";
   1.526 +val Inv_def = thm "Inv_def";
   1.527 +
   1.528 +val someI = thm "someI";
   1.529 +val someI_ex = thm "someI_ex";
   1.530 +val someI2 = thm "someI2";
   1.531 +val someI2_ex = thm "someI2_ex";
   1.532 +val some_equality = thm "some_equality";
   1.533 +val some1_equality = thm "some1_equality";
   1.534 +val some_eq_ex = thm "some_eq_ex";
   1.535 +val some_eq_trivial = thm "some_eq_trivial";
   1.536 +val some_sym_eq_trivial = thm "some_sym_eq_trivial";
   1.537 +val choice = thm "choice";
   1.538 +val bchoice = thm "bchoice";
   1.539 +val inv_id = thm "inv_id";
   1.540 +val inv_f_f = thm "inv_f_f";
   1.541 +val inv_f_eq = thm "inv_f_eq";
   1.542 +val inj_imp_inv_eq = thm "inj_imp_inv_eq";
   1.543 +val inj_transfer = thm "inj_transfer";
   1.544 +val inj_iff = thm "inj_iff";
   1.545 +val inj_imp_surj_inv = thm "inj_imp_surj_inv";
   1.546 +val f_inv_f = thm "f_inv_f";
   1.547 +val surj_f_inv_f = thm "surj_f_inv_f";
   1.548 +val inv_injective = thm "inv_injective";
   1.549 +val inj_on_inv = thm "inj_on_inv";
   1.550 +val surj_imp_inj_inv = thm "surj_imp_inj_inv";
   1.551 +val surj_iff = thm "surj_iff";
   1.552 +val surj_imp_inv_eq = thm "surj_imp_inv_eq";
   1.553 +val bij_imp_bij_inv = thm "bij_imp_bij_inv";
   1.554 +val inv_equality = thm "inv_equality";
   1.555 +val inv_inv_eq = thm "inv_inv_eq";
   1.556 +val o_inv_distrib = thm "o_inv_distrib";
   1.557 +val image_surj_f_inv_f = thm "image_surj_f_inv_f";
   1.558 +val image_inv_f_f = thm "image_inv_f_f";
   1.559 +val inv_image_comp = thm "inv_image_comp";
   1.560 +val bij_image_Collect_eq = thm "bij_image_Collect_eq";
   1.561 +val bij_vimage_eq_inv_image = thm "bij_vimage_eq_inv_image";
   1.562 +val Inv_f_f = thm "Inv_f_f";
   1.563 +val f_Inv_f = thm "f_Inv_f";
   1.564 +val Inv_injective = thm "Inv_injective";
   1.565 +val inj_on_Inv = thm "inj_on_Inv";
   1.566 +val split_paired_Eps = thm "split_paired_Eps";
   1.567 +val Eps_split = thm "Eps_split";
   1.568 +val Eps_split_eq = thm "Eps_split_eq";
   1.569 +val wf_iff_no_infinite_down_chain = thm "wf_iff_no_infinite_down_chain";
   1.570 +val Inv_mem = thm "Inv_mem";
   1.571 +val Inv_f_eq = thm "Inv_f_eq";
   1.572 +val Inv_comp = thm "Inv_comp";
   1.573 +val tfl_some = thm "tfl_some";
   1.574 +val make_neg_rule = thm "make_neg_rule";
   1.575 +val make_refined_neg_rule = thm "make_refined_neg_rule";
   1.576 +val make_pos_rule = thm "make_pos_rule";
   1.577 +val make_neg_rule' = thm "make_neg_rule'";
   1.578 +val make_pos_rule' = thm "make_pos_rule'";
   1.579 +val make_neg_goal = thm "make_neg_goal";
   1.580 +val make_pos_goal = thm "make_pos_goal";
   1.581 +val conj_forward = thm "conj_forward";
   1.582 +val disj_forward = thm "disj_forward";
   1.583 +val disj_forward2 = thm "disj_forward2";
   1.584 +val all_forward = thm "all_forward";
   1.585 +val ex_forward = thm "ex_forward";
   1.586 +*}
   1.587 +
   1.588 +
   1.589  use "Tools/meson.ML"
   1.590  setup meson_setup
   1.591