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
2.1 --- a/src/HOL/Hilbert_Choice_lemmas.ML Wed May 19 11:24:54 2004 +0200
2.2 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000
2.3 @@ -1,281 +0,0 @@
2.4 -(* Title: HOL/Hilbert_Choice_lemmas
2.5 - ID: $Id$
2.6 - Author: Lawrence C Paulson
2.7 - Copyright 2001 University of Cambridge
2.8 -
2.9 -Lemmas for Hilbert's epsilon-operator and the Axiom of Choice
2.10 -*)
2.11 -
2.12 -
2.13 -(* ML bindings *)
2.14 -val someI = thm "someI";
2.15 -
2.16 -section "SOME: Hilbert's Epsilon-operator";
2.17 -
2.18 -(*Easier to apply than someI if witness ?a comes from an EX-formula*)
2.19 -Goal "EX x. P x ==> P (SOME x. P x)";
2.20 -by (etac exE 1);
2.21 -by (etac someI 1);
2.22 -qed "someI_ex";
2.23 -
2.24 -(*Easier to apply than someI: conclusion has only one occurrence of P*)
2.25 -val prems = Goal "[| P a; !!x. P x ==> Q x |] ==> Q (SOME x. P x)";
2.26 -by (resolve_tac prems 1);
2.27 -by (rtac someI 1);
2.28 -by (resolve_tac prems 1) ;
2.29 -qed "someI2";
2.30 -
2.31 -(*Easier to apply than someI2 if witness ?a comes from an EX-formula*)
2.32 -val [major,minor] = Goal "[| EX a. P a; !!x. P x ==> Q x |] ==> Q (SOME x. P x)";
2.33 -by (rtac (major RS exE) 1);
2.34 -by (etac someI2 1 THEN etac minor 1);
2.35 -qed "someI2_ex";
2.36 -
2.37 -val prems = Goal "[| P a; !!x. P x ==> x=a |] ==> (SOME x. P x) = a";
2.38 -by (rtac someI2 1);
2.39 -by (REPEAT (ares_tac prems 1)) ;
2.40 -qed "some_equality";
2.41 -AddIs [some_equality];
2.42 -
2.43 -Goal "[| EX!x. P x; P a |] ==> (SOME x. P x) = a";
2.44 -by (rtac some_equality 1);
2.45 -by (atac 1);
2.46 -by (etac ex1E 1);
2.47 -by (etac all_dupE 1);
2.48 -by (dtac mp 1);
2.49 -by (atac 1);
2.50 -by (etac ssubst 1);
2.51 -by (etac allE 1);
2.52 -by (etac mp 1);
2.53 -by (atac 1);
2.54 -qed "some1_equality";
2.55 -
2.56 -Goal "P (SOME x. P x) = (EX x. P x)";
2.57 -by (rtac iffI 1);
2.58 -by (etac exI 1);
2.59 -by (etac exE 1);
2.60 -by (etac someI 1);
2.61 -qed "some_eq_ex";
2.62 -
2.63 -Goal "(SOME y. y=x) = x";
2.64 -by (rtac some_equality 1);
2.65 -by (rtac refl 1);
2.66 -by (atac 1);
2.67 -qed "some_eq_trivial";
2.68 -
2.69 -Goal "(SOME y. x=y) = x";
2.70 -by (rtac some_equality 1);
2.71 -by (rtac refl 1);
2.72 -by (etac sym 1);
2.73 -qed "some_sym_eq_trivial";
2.74 -Addsimps [some_eq_trivial, some_sym_eq_trivial];
2.75 -
2.76 -
2.77 -(** "Axiom" of Choice, proved using the description operator **)
2.78 -
2.79 -(*Used in Tools/meson.ML*)
2.80 -Goal "ALL x. EX y. Q x y ==> EX f. ALL x. Q x (f x)";
2.81 -by (fast_tac (claset() addEs [someI]) 1);
2.82 -qed "choice";
2.83 -
2.84 -Goal "ALL x:S. EX y. Q x y ==> EX f. ALL x:S. Q x (f x)";
2.85 -by (fast_tac (claset() addEs [someI]) 1);
2.86 -qed "bchoice";
2.87 -
2.88 -
2.89 -section "Function Inverse";
2.90 -
2.91 -val inv_def = thm "inv_def";
2.92 -val Inv_def = thm "Inv_def";
2.93 -
2.94 -Goal "inv id = id";
2.95 -by (simp_tac (simpset() addsimps [inv_def,id_def]) 1);
2.96 -qed "inv_id";
2.97 -Addsimps [inv_id];
2.98 -
2.99 -(*A one-to-one function has an inverse.*)
2.100 -Goalw [inv_def] "inj(f) ==> inv f (f x) = x";
2.101 -by (asm_simp_tac (simpset() addsimps [inj_eq]) 1);
2.102 -qed "inv_f_f";
2.103 -Addsimps [inv_f_f];
2.104 -
2.105 -Goal "[| inj(f); f x = y |] ==> inv f y = x";
2.106 -by (etac subst 1);
2.107 -by (etac inv_f_f 1);
2.108 -qed "inv_f_eq";
2.109 -
2.110 -Goal "[| inj f; ALL x. f(g x) = x |] ==> inv f = g";
2.111 -by (blast_tac (claset() addIs [ext, inv_f_eq]) 1);
2.112 -qed "inj_imp_inv_eq";
2.113 -
2.114 -(* Useful??? *)
2.115 -val [oneone,minor] = Goal
2.116 - "[| inj(f); !!y. y: range(f) ==> P(inv f y) |] ==> P(x)";
2.117 -by (res_inst_tac [("t", "x")] (oneone RS (inv_f_f RS subst)) 1);
2.118 -by (rtac (rangeI RS minor) 1);
2.119 -qed "inj_transfer";
2.120 -
2.121 -Goal "(inj f) = (inv f o f = id)";
2.122 -by (asm_simp_tac (simpset() addsimps [o_def, expand_fun_eq]) 1);
2.123 -by (blast_tac (claset() addIs [inj_inverseI, inv_f_f]) 1);
2.124 -qed "inj_iff";
2.125 -
2.126 -Goal "inj f ==> surj (inv f)";
2.127 -by (blast_tac (claset() addIs [surjI, inv_f_f]) 1);
2.128 -qed "inj_imp_surj_inv";
2.129 -
2.130 -Goalw [inv_def] "y : range(f) ==> f(inv f y) = y";
2.131 -by (fast_tac (claset() addIs [someI]) 1);
2.132 -qed "f_inv_f";
2.133 -
2.134 -Goal "surj f ==> f(inv f y) = y";
2.135 -by (asm_simp_tac (simpset() addsimps [f_inv_f, surj_range]) 1);
2.136 -qed "surj_f_inv_f";
2.137 -
2.138 -Goal "[| inv f x = inv f y; x: range(f); y: range(f) |] ==> x=y";
2.139 -by (rtac (arg_cong RS box_equals) 1);
2.140 -by (REPEAT (ares_tac [f_inv_f] 1));
2.141 -qed "inv_injective";
2.142 -
2.143 -Goal "A <= range(f) ==> inj_on (inv f) A";
2.144 -by (fast_tac (claset() addIs [inj_onI]
2.145 - addEs [inv_injective, injD]) 1);
2.146 -qed "inj_on_inv";
2.147 -
2.148 -Goal "surj f ==> inj (inv f)";
2.149 -by (asm_simp_tac (simpset() addsimps [inj_on_inv, surj_range]) 1);
2.150 -qed "surj_imp_inj_inv";
2.151 -
2.152 -Goal "(surj f) = (f o inv f = id)";
2.153 -by (asm_simp_tac (simpset() addsimps [o_def, expand_fun_eq]) 1);
2.154 -by (blast_tac (claset() addIs [surjI, surj_f_inv_f]) 1);
2.155 -qed "surj_iff";
2.156 -
2.157 -Goal "[| surj f; ALL x. g(f x) = x |] ==> inv f = g";
2.158 -by (rtac ext 1);
2.159 -by (dres_inst_tac [("x","inv f x")] spec 1);
2.160 -by (asm_full_simp_tac (simpset() addsimps [surj_f_inv_f]) 1);
2.161 -qed "surj_imp_inv_eq";
2.162 -
2.163 -Goalw [bij_def] "bij f ==> bij (inv f)";
2.164 -by (asm_simp_tac (simpset() addsimps [inj_imp_surj_inv, surj_imp_inj_inv]) 1);
2.165 -qed "bij_imp_bij_inv";
2.166 -
2.167 -val prems =
2.168 -Goalw [inv_def] "[| !! x. g (f x) = x; !! y. f (g y) = y |] ==> inv f = g";
2.169 -by (rtac ext 1);
2.170 -by (auto_tac (claset(), simpset() addsimps prems));
2.171 -qed "inv_equality";
2.172 -
2.173 -Goalw [bij_def] "bij f ==> inv (inv f) = f";
2.174 -by (rtac inv_equality 1);
2.175 -by (auto_tac (claset(), simpset() addsimps [surj_f_inv_f]));
2.176 -qed "inv_inv_eq";
2.177 -
2.178 -(** bij(inv f) implies little about f. Consider f::bool=>bool such that
2.179 - f(True)=f(False)=True. Then it's consistent with axiom someI that
2.180 - inv(f) could be any function at all, including the identity function.
2.181 - If inv(f)=id then inv(f) is a bijection, but inj(f), surj(f) and
2.182 - inv(inv(f))=f all fail.
2.183 -**)
2.184 -
2.185 -Goalw [bij_def] "[| bij f; bij g |] ==> inv (f o g) = inv g o inv f";
2.186 -by (rtac (inv_equality) 1);
2.187 -by (auto_tac (claset(), simpset() addsimps [surj_f_inv_f]));
2.188 -qed "o_inv_distrib";
2.189 -
2.190 -
2.191 -Goal "surj f ==> f ` (inv f ` A) = A";
2.192 -by (asm_simp_tac (simpset() addsimps [image_eq_UN, surj_f_inv_f]) 1);
2.193 -qed "image_surj_f_inv_f";
2.194 -
2.195 -Goal "inj f ==> (inv f) ` (f ` A) = A";
2.196 -by (asm_simp_tac (simpset() addsimps [image_eq_UN]) 1);
2.197 -qed "image_inv_f_f";
2.198 -
2.199 -Goalw [image_def] "inj(f) ==> inv(f)`(f`X) = X";
2.200 -by Auto_tac;
2.201 -qed "inv_image_comp";
2.202 -
2.203 -Goal "bij f ==> f ` Collect P = {y. P (inv f y)}";
2.204 -by Auto_tac;
2.205 -by (force_tac (claset(), simpset() addsimps [bij_is_inj]) 1);
2.206 -by (blast_tac (claset() addIs [bij_is_surj RS surj_f_inv_f RS sym]) 1);
2.207 -qed "bij_image_Collect_eq";
2.208 -
2.209 -Goal "bij f ==> f -` A = inv f ` A";
2.210 -by Safe_tac;
2.211 -by (asm_simp_tac (simpset() addsimps [bij_is_surj RS surj_f_inv_f]) 2);
2.212 -by (blast_tac (claset() addIs [bij_is_inj RS inv_f_f RS sym]) 1);
2.213 -qed "bij_vimage_eq_inv_image";
2.214 -
2.215 -
2.216 -section "Inverse of a PI-function (restricted domain)";
2.217 -
2.218 -Goal "[| inj_on f A; x : A |] ==> Inv A f (f x) = x";
2.219 -by (asm_full_simp_tac (simpset() addsimps [Inv_def, inj_on_def]) 1);
2.220 -by (blast_tac (claset() addIs [someI2]) 1);
2.221 -qed "Inv_f_f";
2.222 -
2.223 -Goal "y : f`A ==> f (Inv A f y) = y";
2.224 -by (asm_simp_tac (simpset() addsimps [Inv_def]) 1);
2.225 -by (fast_tac (claset() addIs [someI2]) 1);
2.226 -qed "f_Inv_f";
2.227 -
2.228 -Goal "[| Inv A f x = Inv A f y; x : f`A; y : f`A |] ==> x=y";
2.229 -by (rtac (arg_cong RS box_equals) 1);
2.230 -by (REPEAT (ares_tac [f_Inv_f] 1));
2.231 -qed "Inv_injective";
2.232 -
2.233 -Goal "B <= f`A ==> inj_on (Inv A f) B";
2.234 -by (rtac inj_onI 1);
2.235 -by (blast_tac (claset() addIs [inj_onI] addDs [Inv_injective, injD]) 1);
2.236 -qed "inj_on_Inv";
2.237 -
2.238 -
2.239 -
2.240 -section "split and SOME";
2.241 -
2.242 -(*Can't be added to simpset: loops!*)
2.243 -Goal "(SOME x. P x) = (SOME (a,b). P(a,b))";
2.244 -by (simp_tac (simpset() addsimps [split_Pair_apply]) 1);
2.245 -qed "split_paired_Eps";
2.246 -
2.247 -Goalw [split_def] "Eps (split P) = (SOME xy. P (fst xy) (snd xy))";
2.248 -by (rtac refl 1);
2.249 -qed "Eps_split";
2.250 -
2.251 -Goal "(@(x',y'). x = x' & y = y') = (x,y)";
2.252 -by (Blast_tac 1);
2.253 -qed "Eps_split_eq";
2.254 -Addsimps [Eps_split_eq];
2.255 -
2.256 -
2.257 -section "A relation is wellfounded iff it has no infinite descending chain";
2.258 -
2.259 -Goalw [wf_eq_minimal RS eq_reflection]
2.260 - "wf r = (~(EX f. ALL i. (f(Suc i),f i) : r))";
2.261 -by (rtac iffI 1);
2.262 - by (rtac notI 1);
2.263 - by (etac exE 1);
2.264 - by (eres_inst_tac [("x","{w. EX i. w=f i}")] allE 1);
2.265 - by (Blast_tac 1);
2.266 -by (etac contrapos_np 1);
2.267 -by (Asm_full_simp_tac 1);
2.268 -by (Clarify_tac 1);
2.269 -by (subgoal_tac "ALL n. nat_rec x (%i y. @z. z:Q & (z,y):r) n : Q" 1);
2.270 - by (res_inst_tac[("x","nat_rec x (%i y. @z. z:Q & (z,y):r)")]exI 1);
2.271 - by (rtac allI 1);
2.272 - by (Simp_tac 1);
2.273 - by (rtac someI2_ex 1);
2.274 - by (Blast_tac 1);
2.275 - by (Blast_tac 1);
2.276 -by (rtac allI 1);
2.277 -by (induct_tac "n" 1);
2.278 - by (Asm_simp_tac 1);
2.279 -by (Simp_tac 1);
2.280 -by (rtac someI2_ex 1);
2.281 - by (Blast_tac 1);
2.282 -by (Blast_tac 1);
2.283 -qed "wf_iff_no_infinite_down_chain";
2.284 -
3.1 --- a/src/HOL/IsaMakefile Wed May 19 11:24:54 2004 +0200
3.2 +++ b/src/HOL/IsaMakefile Wed May 19 11:29:47 2004 +0200
3.3 @@ -82,8 +82,7 @@
3.4 $(SRC)/TFL/usyntax.ML $(SRC)/TFL/utils.ML \
3.5 Datatype.thy Datatype_Universe.ML Datatype_Universe.thy \
3.6 Divides.thy Extraction.thy Finite_Set.ML Finite_Set.thy \
3.7 - Fun.thy Gfp.ML Gfp.thy \
3.8 - Hilbert_Choice.thy Hilbert_Choice_lemmas.ML HOL.ML \
3.9 + Fun.thy Gfp.ML Gfp.thy Hilbert_Choice.thy HOL.ML \
3.10 HOL.thy HOL_lemmas.ML Inductive.thy Infinite_Set.thy Integ/Bin.thy \
3.11 Integ/cooper_dec.ML Integ/cooper_proof.ML \
3.12 Integ/Equiv.thy Integ/IntArith.thy Integ/IntDef.thy \
3.13 @@ -114,7 +113,7 @@
3.14 Transitive_Closure.thy Transitive_Closure.ML Typedef.thy \
3.15 Wellfounded_Recursion.ML Wellfounded_Recursion.thy Wellfounded_Relations.ML \
3.16 Wellfounded_Relations.thy arith_data.ML blastdata.ML cladata.ML \
3.17 - document/root.tex hologic.ML meson_lemmas.ML simpdata.ML thy_syntax.ML
3.18 + document/root.tex hologic.ML simpdata.ML thy_syntax.ML
3.19 @$(ISATOOL) usedir -b -g true $(HOL_PROOF_OBJECTS) $(OUT)/Pure HOL
3.20
3.21
4.1 --- a/src/HOL/meson_lemmas.ML Wed May 19 11:24:54 2004 +0200
4.2 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000
4.3 @@ -1,95 +0,0 @@
4.4 -(* Title: HOL/meson_lemmas.ML
4.5 - ID: $Id$
4.6 - Author: Lawrence C Paulson, Cambridge University Computer Laboratory
4.7 - Copyright 1992 University of Cambridge
4.8 -
4.9 -Lemmas for Meson.
4.10 -*)
4.11 -
4.12 -(* Generation of contrapositives *)
4.13 -
4.14 -(*Inserts negated disjunct after removing the negation; P is a literal.
4.15 - Model elimination requires assuming the negation of every attempted subgoal,
4.16 - hence the negated disjuncts.*)
4.17 -val [major,minor] = Goal "~P|Q ==> ((~P==>P) ==> Q)";
4.18 -by (rtac (major RS disjE) 1);
4.19 -by (rtac notE 1);
4.20 -by (etac minor 2);
4.21 -by (ALLGOALS assume_tac);
4.22 -qed "make_neg_rule";
4.23 -
4.24 -(*For Plaisted's "Postive refinement" of the MESON procedure*)
4.25 -Goal "~P|Q ==> (P ==> Q)";
4.26 -by (Blast_tac 1);
4.27 -qed "make_refined_neg_rule";
4.28 -
4.29 -(*P should be a literal*)
4.30 -val [major,minor] = Goal "P|Q ==> ((P==>~P) ==> Q)";
4.31 -by (rtac (major RS disjE) 1);
4.32 -by (rtac notE 1);
4.33 -by (etac minor 1);
4.34 -by (ALLGOALS assume_tac);
4.35 -qed "make_pos_rule";
4.36 -
4.37 -(** Versions of make_neg_rule and make_pos_rule that don't insert new
4.38 - assumptions, for ordinary resolution. **)
4.39 -
4.40 -val make_neg_rule' = make_refined_neg_rule;
4.41 -
4.42 -Goal "[|P|Q; ~P|] ==> Q";
4.43 -by (Blast_tac 1);
4.44 -qed "make_pos_rule'";
4.45 -
4.46 -(* Generation of a goal clause -- put away the final literal *)
4.47 -
4.48 -val [major,minor] = Goal "~P ==> ((~P==>P) ==> False)";
4.49 -by (rtac notE 1);
4.50 -by (rtac minor 2);
4.51 -by (ALLGOALS (rtac major));
4.52 -qed "make_neg_goal";
4.53 -
4.54 -val [major,minor] = Goal "P ==> ((P==>~P) ==> False)";
4.55 -by (rtac notE 1);
4.56 -by (rtac minor 1);
4.57 -by (ALLGOALS (rtac major));
4.58 -qed "make_pos_goal";
4.59 -
4.60 -
4.61 -(* Lemmas for forward proof (like congruence rules) *)
4.62 -
4.63 -(*NOTE: could handle conjunctions (faster?) by
4.64 - nf(th RS conjunct2) RS (nf(th RS conjunct1) RS conjI) *)
4.65 -val major::prems = Goal
4.66 - "[| P'&Q'; P' ==> P; Q' ==> Q |] ==> P&Q";
4.67 -by (rtac (major RS conjE) 1);
4.68 -by (rtac conjI 1);
4.69 -by (ALLGOALS (eresolve_tac prems));
4.70 -qed "conj_forward";
4.71 -
4.72 -val major::prems = Goal
4.73 - "[| P'|Q'; P' ==> P; Q' ==> Q |] ==> P|Q";
4.74 -by (rtac (major RS disjE) 1);
4.75 -by (ALLGOALS (dresolve_tac prems));
4.76 -by (ALLGOALS (eresolve_tac [disjI1,disjI2]));
4.77 -qed "disj_forward";
4.78 -
4.79 -(*Version for removal of duplicate literals*)
4.80 -val major::prems = Goal
4.81 - "[| P'|Q'; P' ==> P; [| Q'; P==>False |] ==> Q |] ==> P|Q";
4.82 -by (cut_facts_tac [major] 1);
4.83 -by (blast_tac (claset() addIs prems) 1);
4.84 -qed "disj_forward2";
4.85 -
4.86 -val major::prems = Goal
4.87 - "[| ALL x. P'(x); !!x. P'(x) ==> P(x) |] ==> ALL x. P(x)";
4.88 -by (rtac allI 1);
4.89 -by (resolve_tac prems 1);
4.90 -by (rtac (major RS spec) 1);
4.91 -qed "all_forward";
4.92 -
4.93 -val major::prems = Goal
4.94 - "[| EX x. P'(x); !!x. P'(x) ==> P(x) |] ==> EX x. P(x)";
4.95 -by (rtac (major RS exE) 1);
4.96 -by (rtac exI 1);
4.97 -by (eresolve_tac prems 1);
4.98 -qed "ex_forward";