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+ − 1 (* Title: Nominal2_Base
+ − 2 Authors: Brian Huffman, Christian Urban
+ − 3
+ − 4 Basic definitions and lemma infrastructure for
+ − 5 Nominal Isabelle.
+ − 6 *)
+ − 7 theory Nominal2_Base
+ − 8 imports Main Infinite_Set
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+ − 9 "~~/src/HOL/Quotient_Examples/FSet"
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+ − 10 uses ("nominal_library.ML")
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+ − 11 ("nominal_atoms.ML")
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+ − 12 begin
+ − 13
+ − 14 section {* Atoms and Sorts *}
+ − 15
+ − 16 text {* A simple implementation for atom_sorts is strings. *}
+ − 17 (* types atom_sort = string *)
+ − 18
+ − 19 text {* To deal with Church-like binding we use trees of
+ − 20 strings as sorts. *}
+ − 21
+ − 22 datatype atom_sort = Sort "string" "atom_sort list"
+ − 23
+ − 24 datatype atom = Atom atom_sort nat
+ − 25
+ − 26
+ − 27 text {* Basic projection function. *}
+ − 28
+ − 29 primrec
+ − 30 sort_of :: "atom \<Rightarrow> atom_sort"
+ − 31 where
+ − 32 "sort_of (Atom s i) = s"
+ − 33
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+ − 34 primrec
+ − 35 nat_of :: "atom \<Rightarrow> nat"
+ − 36 where
+ − 37 "nat_of (Atom s n) = n"
+ − 38
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+ − 39
+ − 40 text {* There are infinitely many atoms of each sort. *}
+ − 41 lemma INFM_sort_of_eq:
+ − 42 shows "INFM a. sort_of a = s"
+ − 43 proof -
+ − 44 have "INFM i. sort_of (Atom s i) = s" by simp
+ − 45 moreover have "inj (Atom s)" by (simp add: inj_on_def)
+ − 46 ultimately show "INFM a. sort_of a = s" by (rule INFM_inj)
+ − 47 qed
+ − 48
+ − 49 lemma infinite_sort_of_eq:
+ − 50 shows "infinite {a. sort_of a = s}"
+ − 51 using INFM_sort_of_eq unfolding INFM_iff_infinite .
+ − 52
+ − 53 lemma atom_infinite [simp]:
+ − 54 shows "infinite (UNIV :: atom set)"
+ − 55 using subset_UNIV infinite_sort_of_eq
+ − 56 by (rule infinite_super)
+ − 57
+ − 58 lemma obtain_atom:
+ − 59 fixes X :: "atom set"
+ − 60 assumes X: "finite X"
+ − 61 obtains a where "a \<notin> X" "sort_of a = s"
+ − 62 proof -
+ − 63 from X have "MOST a. a \<notin> X"
+ − 64 unfolding MOST_iff_cofinite by simp
+ − 65 with INFM_sort_of_eq
+ − 66 have "INFM a. sort_of a = s \<and> a \<notin> X"
+ − 67 by (rule INFM_conjI)
+ − 68 then obtain a where "a \<notin> X" "sort_of a = s"
+ − 69 by (auto elim: INFM_E)
+ − 70 then show ?thesis ..
+ − 71 qed
+ − 72
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+ − 73 lemma atom_components_eq_iff:
+ − 74 fixes a b :: atom
+ − 75 shows "a = b \<longleftrightarrow> sort_of a = sort_of b \<and> nat_of a = nat_of b"
+ − 76 by (induct a, induct b, simp)
+ − 77
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+ − 78 section {* Sort-Respecting Permutations *}
+ − 79
+ − 80 typedef perm =
+ − 81 "{f. bij f \<and> finite {a. f a \<noteq> a} \<and> (\<forall>a. sort_of (f a) = sort_of a)}"
+ − 82 proof
+ − 83 show "id \<in> ?perm" by simp
+ − 84 qed
+ − 85
+ − 86 lemma permI:
+ − 87 assumes "bij f" and "MOST x. f x = x" and "\<And>a. sort_of (f a) = sort_of a"
+ − 88 shows "f \<in> perm"
+ − 89 using assms unfolding perm_def MOST_iff_cofinite by simp
+ − 90
+ − 91 lemma perm_is_bij: "f \<in> perm \<Longrightarrow> bij f"
+ − 92 unfolding perm_def by simp
+ − 93
+ − 94 lemma perm_is_finite: "f \<in> perm \<Longrightarrow> finite {a. f a \<noteq> a}"
+ − 95 unfolding perm_def by simp
+ − 96
+ − 97 lemma perm_is_sort_respecting: "f \<in> perm \<Longrightarrow> sort_of (f a) = sort_of a"
+ − 98 unfolding perm_def by simp
+ − 99
+ − 100 lemma perm_MOST: "f \<in> perm \<Longrightarrow> MOST x. f x = x"
+ − 101 unfolding perm_def MOST_iff_cofinite by simp
+ − 102
+ − 103 lemma perm_id: "id \<in> perm"
+ − 104 unfolding perm_def by simp
+ − 105
+ − 106 lemma perm_comp:
+ − 107 assumes f: "f \<in> perm" and g: "g \<in> perm"
+ − 108 shows "(f \<circ> g) \<in> perm"
+ − 109 apply (rule permI)
+ − 110 apply (rule bij_comp)
+ − 111 apply (rule perm_is_bij [OF g])
+ − 112 apply (rule perm_is_bij [OF f])
+ − 113 apply (rule MOST_rev_mp [OF perm_MOST [OF g]])
+ − 114 apply (rule MOST_rev_mp [OF perm_MOST [OF f]])
+ − 115 apply (simp)
+ − 116 apply (simp add: perm_is_sort_respecting [OF f])
+ − 117 apply (simp add: perm_is_sort_respecting [OF g])
+ − 118 done
+ − 119
+ − 120 lemma perm_inv:
+ − 121 assumes f: "f \<in> perm"
+ − 122 shows "(inv f) \<in> perm"
+ − 123 apply (rule permI)
+ − 124 apply (rule bij_imp_bij_inv)
+ − 125 apply (rule perm_is_bij [OF f])
+ − 126 apply (rule MOST_mono [OF perm_MOST [OF f]])
+ − 127 apply (erule subst, rule inv_f_f)
+ − 128 apply (rule bij_is_inj [OF perm_is_bij [OF f]])
+ − 129 apply (rule perm_is_sort_respecting [OF f, THEN sym, THEN trans])
+ − 130 apply (simp add: surj_f_inv_f [OF bij_is_surj [OF perm_is_bij [OF f]]])
+ − 131 done
+ − 132
+ − 133 lemma bij_Rep_perm: "bij (Rep_perm p)"
+ − 134 using Rep_perm [of p] unfolding perm_def by simp
+ − 135
+ − 136 lemma finite_Rep_perm: "finite {a. Rep_perm p a \<noteq> a}"
+ − 137 using Rep_perm [of p] unfolding perm_def by simp
+ − 138
+ − 139 lemma sort_of_Rep_perm: "sort_of (Rep_perm p a) = sort_of a"
+ − 140 using Rep_perm [of p] unfolding perm_def by simp
+ − 141
+ − 142 lemma Rep_perm_ext:
+ − 143 "Rep_perm p1 = Rep_perm p2 \<Longrightarrow> p1 = p2"
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+ − 144 by (simp add: fun_eq_iff Rep_perm_inject [symmetric])
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+ − 145
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+ − 146 instance perm :: size ..
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+ − 147
+ − 148 subsection {* Permutations form a group *}
+ − 149
+ − 150 instantiation perm :: group_add
+ − 151 begin
+ − 152
+ − 153 definition
+ − 154 "0 = Abs_perm id"
+ − 155
+ − 156 definition
+ − 157 "- p = Abs_perm (inv (Rep_perm p))"
+ − 158
+ − 159 definition
+ − 160 "p + q = Abs_perm (Rep_perm p \<circ> Rep_perm q)"
+ − 161
+ − 162 definition
+ − 163 "(p1::perm) - p2 = p1 + - p2"
+ − 164
+ − 165 lemma Rep_perm_0: "Rep_perm 0 = id"
+ − 166 unfolding zero_perm_def
+ − 167 by (simp add: Abs_perm_inverse perm_id)
+ − 168
+ − 169 lemma Rep_perm_add:
+ − 170 "Rep_perm (p1 + p2) = Rep_perm p1 \<circ> Rep_perm p2"
+ − 171 unfolding plus_perm_def
+ − 172 by (simp add: Abs_perm_inverse perm_comp Rep_perm)
+ − 173
+ − 174 lemma Rep_perm_uminus:
+ − 175 "Rep_perm (- p) = inv (Rep_perm p)"
+ − 176 unfolding uminus_perm_def
+ − 177 by (simp add: Abs_perm_inverse perm_inv Rep_perm)
+ − 178
+ − 179 instance
+ − 180 apply default
+ − 181 unfolding Rep_perm_inject [symmetric]
+ − 182 unfolding minus_perm_def
+ − 183 unfolding Rep_perm_add
+ − 184 unfolding Rep_perm_uminus
+ − 185 unfolding Rep_perm_0
+ − 186 by (simp_all add: o_assoc inv_o_cancel [OF bij_is_inj [OF bij_Rep_perm]])
+ − 187
+ − 188 end
+ − 189
+ − 190
+ − 191 section {* Implementation of swappings *}
+ − 192
+ − 193 definition
+ − 194 swap :: "atom \<Rightarrow> atom \<Rightarrow> perm" ("'(_ \<rightleftharpoons> _')")
+ − 195 where
+ − 196 "(a \<rightleftharpoons> b) =
+ − 197 Abs_perm (if sort_of a = sort_of b
+ − 198 then (\<lambda>c. if a = c then b else if b = c then a else c)
+ − 199 else id)"
+ − 200
+ − 201 lemma Rep_perm_swap:
+ − 202 "Rep_perm (a \<rightleftharpoons> b) =
+ − 203 (if sort_of a = sort_of b
+ − 204 then (\<lambda>c. if a = c then b else if b = c then a else c)
+ − 205 else id)"
+ − 206 unfolding swap_def
+ − 207 apply (rule Abs_perm_inverse)
+ − 208 apply (rule permI)
+ − 209 apply (auto simp add: bij_def inj_on_def surj_def)[1]
+ − 210 apply (rule MOST_rev_mp [OF MOST_neq(1) [of a]])
+ − 211 apply (rule MOST_rev_mp [OF MOST_neq(1) [of b]])
+ − 212 apply (simp)
+ − 213 apply (simp)
+ − 214 done
+ − 215
+ − 216 lemmas Rep_perm_simps =
+ − 217 Rep_perm_0
+ − 218 Rep_perm_add
+ − 219 Rep_perm_uminus
+ − 220 Rep_perm_swap
+ − 221
+ − 222 lemma swap_different_sorts [simp]:
+ − 223 "sort_of a \<noteq> sort_of b \<Longrightarrow> (a \<rightleftharpoons> b) = 0"
+ − 224 by (rule Rep_perm_ext) (simp add: Rep_perm_simps)
+ − 225
+ − 226 lemma swap_cancel:
+ − 227 "(a \<rightleftharpoons> b) + (a \<rightleftharpoons> b) = 0"
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+ − 228 by (rule Rep_perm_ext)
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+ − 229 (simp add: Rep_perm_simps fun_eq_iff)
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+ − 230
+ − 231 lemma swap_self [simp]:
+ − 232 "(a \<rightleftharpoons> a) = 0"
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+ − 233 by (rule Rep_perm_ext, simp add: Rep_perm_simps fun_eq_iff)
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+ − 234
+ − 235 lemma minus_swap [simp]:
+ − 236 "- (a \<rightleftharpoons> b) = (a \<rightleftharpoons> b)"
+ − 237 by (rule minus_unique [OF swap_cancel])
+ − 238
+ − 239 lemma swap_commute:
+ − 240 "(a \<rightleftharpoons> b) = (b \<rightleftharpoons> a)"
+ − 241 by (rule Rep_perm_ext)
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+ − 242 (simp add: Rep_perm_swap fun_eq_iff)
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+ − 243
+ − 244 lemma swap_triple:
+ − 245 assumes "a \<noteq> b" and "c \<noteq> b"
+ − 246 assumes "sort_of a = sort_of b" "sort_of b = sort_of c"
+ − 247 shows "(a \<rightleftharpoons> c) + (b \<rightleftharpoons> c) + (a \<rightleftharpoons> c) = (a \<rightleftharpoons> b)"
+ − 248 using assms
+ − 249 by (rule_tac Rep_perm_ext)
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+ − 250 (auto simp add: Rep_perm_simps fun_eq_iff)
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+ − 251
+ − 252
+ − 253 section {* Permutation Types *}
+ − 254
+ − 255 text {*
+ − 256 Infix syntax for @{text permute} has higher precedence than
+ − 257 addition, but lower than unary minus.
+ − 258 *}
+ − 259
+ − 260 class pt =
+ − 261 fixes permute :: "perm \<Rightarrow> 'a \<Rightarrow> 'a" ("_ \<bullet> _" [76, 75] 75)
+ − 262 assumes permute_zero [simp]: "0 \<bullet> x = x"
+ − 263 assumes permute_plus [simp]: "(p + q) \<bullet> x = p \<bullet> (q \<bullet> x)"
+ − 264 begin
+ − 265
+ − 266 lemma permute_diff [simp]:
+ − 267 shows "(p - q) \<bullet> x = p \<bullet> - q \<bullet> x"
+ − 268 unfolding diff_minus by simp
+ − 269
+ − 270 lemma permute_minus_cancel [simp]:
+ − 271 shows "p \<bullet> - p \<bullet> x = x"
+ − 272 and "- p \<bullet> p \<bullet> x = x"
+ − 273 unfolding permute_plus [symmetric] by simp_all
+ − 274
+ − 275 lemma permute_swap_cancel [simp]:
+ − 276 shows "(a \<rightleftharpoons> b) \<bullet> (a \<rightleftharpoons> b) \<bullet> x = x"
+ − 277 unfolding permute_plus [symmetric]
+ − 278 by (simp add: swap_cancel)
+ − 279
+ − 280 lemma permute_swap_cancel2 [simp]:
+ − 281 shows "(a \<rightleftharpoons> b) \<bullet> (b \<rightleftharpoons> a) \<bullet> x = x"
+ − 282 unfolding permute_plus [symmetric]
+ − 283 by (simp add: swap_commute)
+ − 284
+ − 285 lemma inj_permute [simp]:
+ − 286 shows "inj (permute p)"
+ − 287 by (rule inj_on_inverseI)
+ − 288 (rule permute_minus_cancel)
+ − 289
+ − 290 lemma surj_permute [simp]:
+ − 291 shows "surj (permute p)"
+ − 292 by (rule surjI, rule permute_minus_cancel)
+ − 293
+ − 294 lemma bij_permute [simp]:
+ − 295 shows "bij (permute p)"
+ − 296 by (rule bijI [OF inj_permute surj_permute])
+ − 297
+ − 298 lemma inv_permute:
+ − 299 shows "inv (permute p) = permute (- p)"
+ − 300 by (rule inv_equality) (simp_all)
+ − 301
+ − 302 lemma permute_minus:
+ − 303 shows "permute (- p) = inv (permute p)"
+ − 304 by (simp add: inv_permute)
+ − 305
+ − 306 lemma permute_eq_iff [simp]:
+ − 307 shows "p \<bullet> x = p \<bullet> y \<longleftrightarrow> x = y"
+ − 308 by (rule inj_permute [THEN inj_eq])
+ − 309
+ − 310 end
+ − 311
+ − 312 subsection {* Permutations for atoms *}
+ − 313
+ − 314 instantiation atom :: pt
+ − 315 begin
+ − 316
+ − 317 definition
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+ − 318 "p \<bullet> a = (Rep_perm p) a"
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+ − 319
+ − 320 instance
+ − 321 apply(default)
+ − 322 apply(simp_all add: permute_atom_def Rep_perm_simps)
+ − 323 done
+ − 324
+ − 325 end
+ − 326
+ − 327 lemma sort_of_permute [simp]:
+ − 328 shows "sort_of (p \<bullet> a) = sort_of a"
+ − 329 unfolding permute_atom_def by (rule sort_of_Rep_perm)
+ − 330
+ − 331 lemma swap_atom:
+ − 332 shows "(a \<rightleftharpoons> b) \<bullet> c =
+ − 333 (if sort_of a = sort_of b
+ − 334 then (if c = a then b else if c = b then a else c) else c)"
+ − 335 unfolding permute_atom_def
+ − 336 by (simp add: Rep_perm_swap)
+ − 337
+ − 338 lemma swap_atom_simps [simp]:
+ − 339 "sort_of a = sort_of b \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> a = b"
+ − 340 "sort_of a = sort_of b \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> b = a"
+ − 341 "c \<noteq> a \<Longrightarrow> c \<noteq> b \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> c = c"
+ − 342 unfolding swap_atom by simp_all
+ − 343
+ − 344 lemma expand_perm_eq:
+ − 345 fixes p q :: "perm"
+ − 346 shows "p = q \<longleftrightarrow> (\<forall>a::atom. p \<bullet> a = q \<bullet> a)"
+ − 347 unfolding permute_atom_def
+ − 348 by (metis Rep_perm_ext ext)
+ − 349
+ − 350
+ − 351 subsection {* Permutations for permutations *}
+ − 352
+ − 353 instantiation perm :: pt
+ − 354 begin
+ − 355
+ − 356 definition
+ − 357 "p \<bullet> q = p + q - p"
+ − 358
+ − 359 instance
+ − 360 apply default
+ − 361 apply (simp add: permute_perm_def)
+ − 362 apply (simp add: permute_perm_def diff_minus minus_add add_assoc)
+ − 363 done
+ − 364
+ − 365 end
+ − 366
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+ − 367 lemma permute_self:
+ − 368 shows "p \<bullet> p = p"
+ − 369 unfolding permute_perm_def
+ − 370 by (simp add: diff_minus add_assoc)
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+ − 371
+ − 372 lemma permute_eqvt:
+ − 373 shows "p \<bullet> (q \<bullet> x) = (p \<bullet> q) \<bullet> (p \<bullet> x)"
+ − 374 unfolding permute_perm_def by simp
+ − 375
+ − 376 lemma zero_perm_eqvt:
+ − 377 shows "p \<bullet> (0::perm) = 0"
+ − 378 unfolding permute_perm_def by simp
+ − 379
+ − 380 lemma add_perm_eqvt:
+ − 381 fixes p p1 p2 :: perm
+ − 382 shows "p \<bullet> (p1 + p2) = p \<bullet> p1 + p \<bullet> p2"
+ − 383 unfolding permute_perm_def
+ − 384 by (simp add: expand_perm_eq)
+ − 385
+ − 386 lemma swap_eqvt:
+ − 387 shows "p \<bullet> (a \<rightleftharpoons> b) = (p \<bullet> a \<rightleftharpoons> p \<bullet> b)"
+ − 388 unfolding permute_perm_def
+ − 389 by (auto simp add: swap_atom expand_perm_eq)
+ − 390
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+ − 391 lemma uminus_eqvt:
+ − 392 fixes p q::"perm"
+ − 393 shows "p \<bullet> (- q) = - (p \<bullet> q)"
+ − 394 unfolding permute_perm_def
+ − 395 by (simp add: diff_minus minus_add add_assoc)
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+ − 396
+ − 397 subsection {* Permutations for functions *}
+ − 398
+ − 399 instantiation "fun" :: (pt, pt) pt
+ − 400 begin
+ − 401
+ − 402 definition
+ − 403 "p \<bullet> f = (\<lambda>x. p \<bullet> (f (- p \<bullet> x)))"
+ − 404
+ − 405 instance
+ − 406 apply default
+ − 407 apply (simp add: permute_fun_def)
+ − 408 apply (simp add: permute_fun_def minus_add)
+ − 409 done
+ − 410
+ − 411 end
+ − 412
+ − 413 lemma permute_fun_app_eq:
+ − 414 shows "p \<bullet> (f x) = (p \<bullet> f) (p \<bullet> x)"
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+ − 415 unfolding permute_fun_def by simp
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+ − 416
+ − 417 subsection {* Permutations for booleans *}
+ − 418
+ − 419 instantiation bool :: pt
+ − 420 begin
+ − 421
+ − 422 definition "p \<bullet> (b::bool) = b"
+ − 423
+ − 424 instance
+ − 425 apply(default)
+ − 426 apply(simp_all add: permute_bool_def)
+ − 427 done
+ − 428
+ − 429 end
+ − 430
+ − 431 lemma Not_eqvt:
+ − 432 shows "p \<bullet> (\<not> A) = (\<not> (p \<bullet> A))"
+ − 433 by (simp add: permute_bool_def)
+ − 434
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+ − 435 lemma conj_eqvt:
+ − 436 shows "p \<bullet> (A \<and> B) = ((p \<bullet> A) \<and> (p \<bullet> B))"
+ − 437 by (simp add: permute_bool_def)
+ − 438
+ − 439 lemma imp_eqvt:
+ − 440 shows "p \<bullet> (A \<longrightarrow> B) = ((p \<bullet> A) \<longrightarrow> (p \<bullet> B))"
+ − 441 by (simp add: permute_bool_def)
+ − 442
+ − 443 lemma ex_eqvt:
+ − 444 shows "p \<bullet> (\<exists>x. P x) = (\<exists>x. (p \<bullet> P) x)"
+ − 445 unfolding permute_fun_def permute_bool_def
+ − 446 by (auto, rule_tac x="p \<bullet> x" in exI, simp)
+ − 447
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+ − 448 lemma all_eqvt:
+ − 449 shows "p \<bullet> (\<forall>x. P x) = (\<forall>x. (p \<bullet> P) x)"
+ − 450 unfolding permute_fun_def permute_bool_def
+ − 451 by (auto, drule_tac x="p \<bullet> x" in spec, simp)
+ − 452
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+ − 453 lemma permute_boolE:
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+ − 454 fixes P::"bool"
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+ − 455 shows "p \<bullet> P \<Longrightarrow> P"
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+ − 456 by (simp add: permute_bool_def)
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+ − 457
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+ − 458 lemma permute_boolI:
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+ − 459 fixes P::"bool"
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+ − 460 shows "P \<Longrightarrow> p \<bullet> P"
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+ − 461 by(simp add: permute_bool_def)
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+ − 462
+ − 463 subsection {* Permutations for sets *}
+ − 464
+ − 465 lemma permute_set_eq:
+ − 466 fixes x::"'a::pt"
+ − 467 and p::"perm"
+ − 468 shows "(p \<bullet> X) = {p \<bullet> x | x. x \<in> X}"
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+ − 469 unfolding permute_fun_def
+ − 470 unfolding permute_bool_def
+ − 471 apply(auto simp add: mem_def)
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+ − 472 apply(rule_tac x="- p \<bullet> x" in exI)
+ − 473 apply(simp)
+ − 474 done
+ − 475
+ − 476 lemma permute_set_eq_image:
+ − 477 shows "p \<bullet> X = permute p ` X"
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+ − 478 unfolding permute_set_eq by auto
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+ − 479
+ − 480 lemma permute_set_eq_vimage:
+ − 481 shows "p \<bullet> X = permute (- p) -` X"
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+ − 482 unfolding permute_fun_def permute_bool_def
+ − 483 unfolding vimage_def Collect_def mem_def ..
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+ − 484
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+ − 485 lemma permute_finite [simp]:
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+ − 486 shows "finite (p \<bullet> X) = finite X"
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+ − 487 apply(simp add: permute_set_eq_image)
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+ − 488 apply(rule iffI)
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+ − 489 apply(drule finite_imageD)
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+ − 490 using inj_permute[where p="p"]
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+ − 491 apply(simp add: inj_on_def)
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+ − 492 apply(assumption)
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+ − 493 apply(rule finite_imageI)
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+ − 494 apply(assumption)
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+ − 495 done
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diff
changeset
+ − 496
1062
+ − 497 lemma swap_set_not_in:
+ − 498 assumes a: "a \<notin> S" "b \<notin> S"
+ − 499 shows "(a \<rightleftharpoons> b) \<bullet> S = S"
1879
+ − 500 unfolding permute_set_eq
+ − 501 using a by (auto simp add: swap_atom)
1062
+ − 502
+ − 503 lemma swap_set_in:
+ − 504 assumes a: "a \<in> S" "b \<notin> S" "sort_of a = sort_of b"
+ − 505 shows "(a \<rightleftharpoons> b) \<bullet> S \<noteq> S"
1879
+ − 506 unfolding permute_set_eq
+ − 507 using a by (auto simp add: swap_atom)
1062
+ − 508
2470
+ − 509 lemma mem_permute_iff:
+ − 510 shows "(p \<bullet> x) \<in> (p \<bullet> X) \<longleftrightarrow> x \<in> X"
+ − 511 unfolding mem_def permute_fun_def permute_bool_def
+ − 512 by simp
+ − 513
+ − 514 lemma mem_eqvt:
+ − 515 shows "p \<bullet> (x \<in> A) \<longleftrightarrow> (p \<bullet> x) \<in> (p \<bullet> A)"
+ − 516 unfolding mem_def
+ − 517 by (simp add: permute_fun_app_eq)
+ − 518
2565
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+ − 519 lemma empty_eqvt:
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+ − 520 shows "p \<bullet> {} = {}"
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changeset
+ − 521 unfolding empty_def Collect_def
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+ − 522 by (simp add: permute_fun_def permute_bool_def)
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diff
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+ − 523
2470
+ − 524 lemma insert_eqvt:
+ − 525 shows "p \<bullet> (insert x A) = insert (p \<bullet> x) (p \<bullet> A)"
+ − 526 unfolding permute_set_eq_image image_insert ..
+ − 527
+ − 528
1062
+ − 529 subsection {* Permutations for units *}
+ − 530
+ − 531 instantiation unit :: pt
+ − 532 begin
+ − 533
+ − 534 definition "p \<bullet> (u::unit) = u"
+ − 535
1879
+ − 536 instance
+ − 537 by (default) (simp_all add: permute_unit_def)
1062
+ − 538
+ − 539 end
+ − 540
+ − 541
+ − 542 subsection {* Permutations for products *}
+ − 543
2378
+ − 544 instantiation prod :: (pt, pt) pt
1062
+ − 545 begin
+ − 546
+ − 547 primrec
+ − 548 permute_prod
+ − 549 where
+ − 550 Pair_eqvt: "p \<bullet> (x, y) = (p \<bullet> x, p \<bullet> y)"
+ − 551
+ − 552 instance
+ − 553 by default auto
+ − 554
+ − 555 end
+ − 556
+ − 557 subsection {* Permutations for sums *}
+ − 558
2378
+ − 559 instantiation sum :: (pt, pt) pt
1062
+ − 560 begin
+ − 561
+ − 562 primrec
+ − 563 permute_sum
+ − 564 where
+ − 565 "p \<bullet> (Inl x) = Inl (p \<bullet> x)"
+ − 566 | "p \<bullet> (Inr y) = Inr (p \<bullet> y)"
+ − 567
1879
+ − 568 instance
+ − 569 by (default) (case_tac [!] x, simp_all)
1062
+ − 570
+ − 571 end
+ − 572
+ − 573 subsection {* Permutations for lists *}
+ − 574
+ − 575 instantiation list :: (pt) pt
+ − 576 begin
+ − 577
+ − 578 primrec
+ − 579 permute_list
+ − 580 where
+ − 581 "p \<bullet> [] = []"
+ − 582 | "p \<bullet> (x # xs) = p \<bullet> x # p \<bullet> xs"
+ − 583
1879
+ − 584 instance
+ − 585 by (default) (induct_tac [!] x, simp_all)
1062
+ − 586
+ − 587 end
+ − 588
2565
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diff
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+ − 589 lemma set_eqvt:
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diff
changeset
+ − 590 shows "p \<bullet> (set xs) = set (p \<bullet> xs)"
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diff
changeset
+ − 591 by (induct xs) (simp_all add: empty_eqvt insert_eqvt)
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diff
changeset
+ − 592
1062
+ − 593 subsection {* Permutations for options *}
+ − 594
+ − 595 instantiation option :: (pt) pt
+ − 596 begin
+ − 597
+ − 598 primrec
+ − 599 permute_option
+ − 600 where
+ − 601 "p \<bullet> None = None"
+ − 602 | "p \<bullet> (Some x) = Some (p \<bullet> x)"
+ − 603
1879
+ − 604 instance
+ − 605 by (default) (induct_tac [!] x, simp_all)
1062
+ − 606
+ − 607 end
+ − 608
2565
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diff
changeset
+ − 609
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diff
changeset
+ − 610 subsection {* Permutations for fsets *}
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diff
changeset
+ − 611
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diff
changeset
+ − 612 lemma permute_fset_rsp[quot_respect]:
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diff
changeset
+ − 613 shows "(op = ===> list_eq ===> list_eq) permute permute"
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diff
changeset
+ − 614 unfolding fun_rel_def
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diff
changeset
+ − 615 by (simp add: set_eqvt[symmetric])
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diff
changeset
+ − 616
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diff
changeset
+ − 617 instantiation fset :: (pt) pt
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diff
changeset
+ − 618 begin
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diff
changeset
+ − 619
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diff
changeset
+ − 620 quotient_definition
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diff
changeset
+ − 621 "permute_fset :: perm \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
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diff
changeset
+ − 622 is
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diff
changeset
+ − 623 "permute :: perm \<Rightarrow> 'a list \<Rightarrow> 'a list"
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diff
changeset
+ − 624
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diff
changeset
+ − 625 instance
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diff
changeset
+ − 626 proof
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diff
changeset
+ − 627 fix x :: "'a fset" and p q :: "perm"
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diff
changeset
+ − 628 show "0 \<bullet> x = x" by (descending) (simp)
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diff
changeset
+ − 629 show "(p + q) \<bullet> x = p \<bullet> q \<bullet> x" by (descending) (simp)
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diff
changeset
+ − 630 qed
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diff
changeset
+ − 631
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diff
changeset
+ − 632 end
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diff
changeset
+ − 633
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diff
changeset
+ − 634 lemma permute_fset [simp]:
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diff
changeset
+ − 635 fixes S::"('a::pt) fset"
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diff
changeset
+ − 636 shows "(p \<bullet> {||}) = ({||} ::('a::pt) fset)"
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diff
changeset
+ − 637 and "(p \<bullet> insert_fset x S) = insert_fset (p \<bullet> x) (p \<bullet> S)"
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diff
changeset
+ − 638 by (lifting permute_list.simps)
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moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 639
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Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 640
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diff
changeset
+ − 641
1062
+ − 642 subsection {* Permutations for @{typ char}, @{typ nat}, and @{typ int} *}
+ − 643
+ − 644 instantiation char :: pt
+ − 645 begin
+ − 646
+ − 647 definition "p \<bullet> (c::char) = c"
+ − 648
1879
+ − 649 instance
+ − 650 by (default) (simp_all add: permute_char_def)
1062
+ − 651
+ − 652 end
+ − 653
+ − 654 instantiation nat :: pt
+ − 655 begin
+ − 656
+ − 657 definition "p \<bullet> (n::nat) = n"
+ − 658
1879
+ − 659 instance
+ − 660 by (default) (simp_all add: permute_nat_def)
1062
+ − 661
+ − 662 end
+ − 663
+ − 664 instantiation int :: pt
+ − 665 begin
+ − 666
+ − 667 definition "p \<bullet> (i::int) = i"
+ − 668
1879
+ − 669 instance
+ − 670 by (default) (simp_all add: permute_int_def)
1062
+ − 671
+ − 672 end
+ − 673
+ − 674
+ − 675 section {* Pure types *}
+ − 676
+ − 677 text {* Pure types will have always empty support. *}
+ − 678
+ − 679 class pure = pt +
+ − 680 assumes permute_pure: "p \<bullet> x = x"
+ − 681
+ − 682 text {* Types @{typ unit} and @{typ bool} are pure. *}
+ − 683
+ − 684 instance unit :: pure
+ − 685 proof qed (rule permute_unit_def)
+ − 686
+ − 687 instance bool :: pure
+ − 688 proof qed (rule permute_bool_def)
+ − 689
+ − 690 text {* Other type constructors preserve purity. *}
+ − 691
+ − 692 instance "fun" :: (pure, pure) pure
+ − 693 by default (simp add: permute_fun_def permute_pure)
+ − 694
2378
+ − 695 instance prod :: (pure, pure) pure
1062
+ − 696 by default (induct_tac x, simp add: permute_pure)
+ − 697
2378
+ − 698 instance sum :: (pure, pure) pure
1062
+ − 699 by default (induct_tac x, simp_all add: permute_pure)
+ − 700
+ − 701 instance list :: (pure) pure
+ − 702 by default (induct_tac x, simp_all add: permute_pure)
+ − 703
+ − 704 instance option :: (pure) pure
+ − 705 by default (induct_tac x, simp_all add: permute_pure)
+ − 706
+ − 707
+ − 708 subsection {* Types @{typ char}, @{typ nat}, and @{typ int} *}
+ − 709
+ − 710 instance char :: pure
+ − 711 proof qed (rule permute_char_def)
+ − 712
+ − 713 instance nat :: pure
+ − 714 proof qed (rule permute_nat_def)
+ − 715
+ − 716 instance int :: pure
+ − 717 proof qed (rule permute_int_def)
+ − 718
+ − 719
+ − 720 subsection {* Supp, Freshness and Supports *}
+ − 721
+ − 722 context pt
+ − 723 begin
+ − 724
+ − 725 definition
+ − 726 supp :: "'a \<Rightarrow> atom set"
+ − 727 where
+ − 728 "supp x = {a. infinite {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x}}"
+ − 729
+ − 730 end
+ − 731
+ − 732 definition
+ − 733 fresh :: "atom \<Rightarrow> 'a::pt \<Rightarrow> bool" ("_ \<sharp> _" [55, 55] 55)
+ − 734 where
+ − 735 "a \<sharp> x \<equiv> a \<notin> supp x"
+ − 736
+ − 737 lemma supp_conv_fresh:
+ − 738 shows "supp x = {a. \<not> a \<sharp> x}"
+ − 739 unfolding fresh_def by simp
+ − 740
+ − 741 lemma swap_rel_trans:
+ − 742 assumes "sort_of a = sort_of b"
+ − 743 assumes "sort_of b = sort_of c"
+ − 744 assumes "(a \<rightleftharpoons> c) \<bullet> x = x"
+ − 745 assumes "(b \<rightleftharpoons> c) \<bullet> x = x"
+ − 746 shows "(a \<rightleftharpoons> b) \<bullet> x = x"
+ − 747 proof (cases)
+ − 748 assume "a = b \<or> c = b"
+ − 749 with assms show "(a \<rightleftharpoons> b) \<bullet> x = x" by auto
+ − 750 next
+ − 751 assume *: "\<not> (a = b \<or> c = b)"
+ − 752 have "((a \<rightleftharpoons> c) + (b \<rightleftharpoons> c) + (a \<rightleftharpoons> c)) \<bullet> x = x"
+ − 753 using assms by simp
+ − 754 also have "(a \<rightleftharpoons> c) + (b \<rightleftharpoons> c) + (a \<rightleftharpoons> c) = (a \<rightleftharpoons> b)"
+ − 755 using assms * by (simp add: swap_triple)
+ − 756 finally show "(a \<rightleftharpoons> b) \<bullet> x = x" .
+ − 757 qed
+ − 758
+ − 759 lemma swap_fresh_fresh:
+ − 760 assumes a: "a \<sharp> x"
+ − 761 and b: "b \<sharp> x"
+ − 762 shows "(a \<rightleftharpoons> b) \<bullet> x = x"
+ − 763 proof (cases)
+ − 764 assume asm: "sort_of a = sort_of b"
+ − 765 have "finite {c. (a \<rightleftharpoons> c) \<bullet> x \<noteq> x}" "finite {c. (b \<rightleftharpoons> c) \<bullet> x \<noteq> x}"
+ − 766 using a b unfolding fresh_def supp_def by simp_all
+ − 767 then have "finite ({c. (a \<rightleftharpoons> c) \<bullet> x \<noteq> x} \<union> {c. (b \<rightleftharpoons> c) \<bullet> x \<noteq> x})" by simp
+ − 768 then obtain c
+ − 769 where "(a \<rightleftharpoons> c) \<bullet> x = x" "(b \<rightleftharpoons> c) \<bullet> x = x" "sort_of c = sort_of b"
+ − 770 by (rule obtain_atom) (auto)
+ − 771 then show "(a \<rightleftharpoons> b) \<bullet> x = x" using asm by (rule_tac swap_rel_trans) (simp_all)
+ − 772 next
+ − 773 assume "sort_of a \<noteq> sort_of b"
+ − 774 then show "(a \<rightleftharpoons> b) \<bullet> x = x" by simp
+ − 775 qed
+ − 776
+ − 777
+ − 778 subsection {* supp and fresh are equivariant *}
+ − 779
+ − 780 lemma finite_Collect_bij:
+ − 781 assumes a: "bij f"
+ − 782 shows "finite {x. P (f x)} = finite {x. P x}"
+ − 783 by (metis a finite_vimage_iff vimage_Collect_eq)
+ − 784
+ − 785 lemma fresh_permute_iff:
+ − 786 shows "(p \<bullet> a) \<sharp> (p \<bullet> x) \<longleftrightarrow> a \<sharp> x"
+ − 787 proof -
+ − 788 have "(p \<bullet> a) \<sharp> (p \<bullet> x) \<longleftrightarrow> finite {b. (p \<bullet> a \<rightleftharpoons> b) \<bullet> p \<bullet> x \<noteq> p \<bullet> x}"
+ − 789 unfolding fresh_def supp_def by simp
+ − 790 also have "\<dots> \<longleftrightarrow> finite {b. (p \<bullet> a \<rightleftharpoons> p \<bullet> b) \<bullet> p \<bullet> x \<noteq> p \<bullet> x}"
1879
+ − 791 using bij_permute by (rule finite_Collect_bij[symmetric])
1062
+ − 792 also have "\<dots> \<longleftrightarrow> finite {b. p \<bullet> (a \<rightleftharpoons> b) \<bullet> x \<noteq> p \<bullet> x}"
+ − 793 by (simp only: permute_eqvt [of p] swap_eqvt)
+ − 794 also have "\<dots> \<longleftrightarrow> finite {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x}"
+ − 795 by (simp only: permute_eq_iff)
+ − 796 also have "\<dots> \<longleftrightarrow> a \<sharp> x"
+ − 797 unfolding fresh_def supp_def by simp
1879
+ − 798 finally show "(p \<bullet> a) \<sharp> (p \<bullet> x) \<longleftrightarrow> a \<sharp> x" .
1062
+ − 799 qed
+ − 800
+ − 801 lemma fresh_eqvt:
+ − 802 shows "p \<bullet> (a \<sharp> x) = (p \<bullet> a) \<sharp> (p \<bullet> x)"
1879
+ − 803 unfolding permute_bool_def
+ − 804 by (simp add: fresh_permute_iff)
1062
+ − 805
+ − 806 lemma supp_eqvt:
+ − 807 fixes p :: "perm"
+ − 808 and x :: "'a::pt"
+ − 809 shows "p \<bullet> (supp x) = supp (p \<bullet> x)"
+ − 810 unfolding supp_conv_fresh
1879
+ − 811 unfolding Collect_def
+ − 812 unfolding permute_fun_def
1062
+ − 813 by (simp add: Not_eqvt fresh_eqvt)
+ − 814
+ − 815 subsection {* supports *}
+ − 816
+ − 817 definition
+ − 818 supports :: "atom set \<Rightarrow> 'a::pt \<Rightarrow> bool" (infixl "supports" 80)
+ − 819 where
+ − 820 "S supports x \<equiv> \<forall>a b. (a \<notin> S \<and> b \<notin> S \<longrightarrow> (a \<rightleftharpoons> b) \<bullet> x = x)"
+ − 821
+ − 822 lemma supp_is_subset:
+ − 823 fixes S :: "atom set"
+ − 824 and x :: "'a::pt"
+ − 825 assumes a1: "S supports x"
+ − 826 and a2: "finite S"
+ − 827 shows "(supp x) \<subseteq> S"
+ − 828 proof (rule ccontr)
1879
+ − 829 assume "\<not> (supp x \<subseteq> S)"
1062
+ − 830 then obtain a where b1: "a \<in> supp x" and b2: "a \<notin> S" by auto
1879
+ − 831 from a1 b2 have "\<forall>b. b \<notin> S \<longrightarrow> (a \<rightleftharpoons> b) \<bullet> x = x" unfolding supports_def by auto
+ − 832 then have "{b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x} \<subseteq> S" by auto
1062
+ − 833 with a2 have "finite {b. (a \<rightleftharpoons> b)\<bullet>x \<noteq> x}" by (simp add: finite_subset)
+ − 834 then have "a \<notin> (supp x)" unfolding supp_def by simp
+ − 835 with b1 show False by simp
+ − 836 qed
+ − 837
+ − 838 lemma supports_finite:
+ − 839 fixes S :: "atom set"
+ − 840 and x :: "'a::pt"
+ − 841 assumes a1: "S supports x"
+ − 842 and a2: "finite S"
+ − 843 shows "finite (supp x)"
+ − 844 proof -
+ − 845 have "(supp x) \<subseteq> S" using a1 a2 by (rule supp_is_subset)
+ − 846 then show "finite (supp x)" using a2 by (simp add: finite_subset)
+ − 847 qed
+ − 848
+ − 849 lemma supp_supports:
+ − 850 fixes x :: "'a::pt"
+ − 851 shows "(supp x) supports x"
1879
+ − 852 unfolding supports_def
+ − 853 proof (intro strip)
1062
+ − 854 fix a b
+ − 855 assume "a \<notin> (supp x) \<and> b \<notin> (supp x)"
+ − 856 then have "a \<sharp> x" and "b \<sharp> x" by (simp_all add: fresh_def)
1879
+ − 857 then show "(a \<rightleftharpoons> b) \<bullet> x = x" by (simp add: swap_fresh_fresh)
1062
+ − 858 qed
+ − 859
+ − 860 lemma supp_is_least_supports:
+ − 861 fixes S :: "atom set"
+ − 862 and x :: "'a::pt"
+ − 863 assumes a1: "S supports x"
+ − 864 and a2: "finite S"
+ − 865 and a3: "\<And>S'. finite S' \<Longrightarrow> (S' supports x) \<Longrightarrow> S \<subseteq> S'"
+ − 866 shows "(supp x) = S"
+ − 867 proof (rule equalityI)
+ − 868 show "(supp x) \<subseteq> S" using a1 a2 by (rule supp_is_subset)
+ − 869 with a2 have fin: "finite (supp x)" by (rule rev_finite_subset)
+ − 870 have "(supp x) supports x" by (rule supp_supports)
+ − 871 with fin a3 show "S \<subseteq> supp x" by blast
+ − 872 qed
+ − 873
+ − 874 lemma subsetCI:
+ − 875 shows "(\<And>x. x \<in> A \<Longrightarrow> x \<notin> B \<Longrightarrow> False) \<Longrightarrow> A \<subseteq> B"
+ − 876 by auto
+ − 877
+ − 878 lemma finite_supp_unique:
+ − 879 assumes a1: "S supports x"
+ − 880 assumes a2: "finite S"
+ − 881 assumes a3: "\<And>a b. \<lbrakk>a \<in> S; b \<notin> S; sort_of a = sort_of b\<rbrakk> \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> x \<noteq> x"
+ − 882 shows "(supp x) = S"
+ − 883 using a1 a2
+ − 884 proof (rule supp_is_least_supports)
+ − 885 fix S'
+ − 886 assume "finite S'" and "S' supports x"
+ − 887 show "S \<subseteq> S'"
+ − 888 proof (rule subsetCI)
+ − 889 fix a
+ − 890 assume "a \<in> S" and "a \<notin> S'"
+ − 891 have "finite (S \<union> S')"
+ − 892 using `finite S` `finite S'` by simp
+ − 893 then obtain b where "b \<notin> S \<union> S'" and "sort_of b = sort_of a"
+ − 894 by (rule obtain_atom)
+ − 895 then have "b \<notin> S" and "b \<notin> S'" and "sort_of a = sort_of b"
+ − 896 by simp_all
+ − 897 then have "(a \<rightleftharpoons> b) \<bullet> x = x"
+ − 898 using `a \<notin> S'` `S' supports x` by (simp add: supports_def)
+ − 899 moreover have "(a \<rightleftharpoons> b) \<bullet> x \<noteq> x"
+ − 900 using `a \<in> S` `b \<notin> S` `sort_of a = sort_of b`
+ − 901 by (rule a3)
+ − 902 ultimately show "False" by simp
+ − 903 qed
+ − 904 qed
+ − 905
2475
486d4647bb37
supp-proofs work except for CoreHaskell and Modules (induct is probably not finding the correct instance)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 906 section {* Support w.r.t. relations *}
486d4647bb37
supp-proofs work except for CoreHaskell and Modules (induct is probably not finding the correct instance)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 907
486d4647bb37
supp-proofs work except for CoreHaskell and Modules (induct is probably not finding the correct instance)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 908 text {*
486d4647bb37
supp-proofs work except for CoreHaskell and Modules (induct is probably not finding the correct instance)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 909 This definition is used for unquotient types, where
486d4647bb37
supp-proofs work except for CoreHaskell and Modules (induct is probably not finding the correct instance)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 910 alpha-equivalence does not coincide with equality.
486d4647bb37
supp-proofs work except for CoreHaskell and Modules (induct is probably not finding the correct instance)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 911 *}
486d4647bb37
supp-proofs work except for CoreHaskell and Modules (induct is probably not finding the correct instance)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 912
486d4647bb37
supp-proofs work except for CoreHaskell and Modules (induct is probably not finding the correct instance)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 913 definition
486d4647bb37
supp-proofs work except for CoreHaskell and Modules (induct is probably not finding the correct instance)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 914 "supp_rel R x = {a. infinite {b. \<not>(R ((a \<rightleftharpoons> b) \<bullet> x) x)}}"
486d4647bb37
supp-proofs work except for CoreHaskell and Modules (induct is probably not finding the correct instance)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 915
486d4647bb37
supp-proofs work except for CoreHaskell and Modules (induct is probably not finding the correct instance)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 916
486d4647bb37
supp-proofs work except for CoreHaskell and Modules (induct is probably not finding the correct instance)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 917
1062
+ − 918 section {* Finitely-supported types *}
+ − 919
+ − 920 class fs = pt +
+ − 921 assumes finite_supp: "finite (supp x)"
+ − 922
+ − 923 lemma pure_supp:
+ − 924 shows "supp (x::'a::pure) = {}"
+ − 925 unfolding supp_def by (simp add: permute_pure)
+ − 926
+ − 927 lemma pure_fresh:
+ − 928 fixes x::"'a::pure"
+ − 929 shows "a \<sharp> x"
+ − 930 unfolding fresh_def by (simp add: pure_supp)
+ − 931
+ − 932 instance pure < fs
+ − 933 by default (simp add: pure_supp)
+ − 934
+ − 935
+ − 936 subsection {* Type @{typ atom} is finitely-supported. *}
+ − 937
+ − 938 lemma supp_atom:
+ − 939 shows "supp a = {a}"
+ − 940 apply (rule finite_supp_unique)
+ − 941 apply (clarsimp simp add: supports_def)
+ − 942 apply simp
+ − 943 apply simp
+ − 944 done
+ − 945
+ − 946 lemma fresh_atom:
+ − 947 shows "a \<sharp> b \<longleftrightarrow> a \<noteq> b"
+ − 948 unfolding fresh_def supp_atom by simp
+ − 949
+ − 950 instance atom :: fs
+ − 951 by default (simp add: supp_atom)
+ − 952
1933
9eab1dfc14d2
moved lemmas from FSet.thy to do with atom to Nominal2_Base, and to do with 'a::at set to Nominal2_Atoms; moved Nominal2_Eqvt.thy one up to be loaded before Nominal2_Atoms
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 953 section {* Support for finite sets of atoms *}
9eab1dfc14d2
moved lemmas from FSet.thy to do with atom to Nominal2_Base, and to do with 'a::at set to Nominal2_Atoms; moved Nominal2_Eqvt.thy one up to be loaded before Nominal2_Atoms
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 954
9eab1dfc14d2
moved lemmas from FSet.thy to do with atom to Nominal2_Base, and to do with 'a::at set to Nominal2_Atoms; moved Nominal2_Eqvt.thy one up to be loaded before Nominal2_Atoms
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 955 lemma supp_finite_atom_set:
9eab1dfc14d2
moved lemmas from FSet.thy to do with atom to Nominal2_Base, and to do with 'a::at set to Nominal2_Atoms; moved Nominal2_Eqvt.thy one up to be loaded before Nominal2_Atoms
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 956 fixes S::"atom set"
9eab1dfc14d2
moved lemmas from FSet.thy to do with atom to Nominal2_Base, and to do with 'a::at set to Nominal2_Atoms; moved Nominal2_Eqvt.thy one up to be loaded before Nominal2_Atoms
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 957 assumes "finite S"
9eab1dfc14d2
moved lemmas from FSet.thy to do with atom to Nominal2_Base, and to do with 'a::at set to Nominal2_Atoms; moved Nominal2_Eqvt.thy one up to be loaded before Nominal2_Atoms
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 958 shows "supp S = S"
9eab1dfc14d2
moved lemmas from FSet.thy to do with atom to Nominal2_Base, and to do with 'a::at set to Nominal2_Atoms; moved Nominal2_Eqvt.thy one up to be loaded before Nominal2_Atoms
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 959 apply(rule finite_supp_unique)
9eab1dfc14d2
moved lemmas from FSet.thy to do with atom to Nominal2_Base, and to do with 'a::at set to Nominal2_Atoms; moved Nominal2_Eqvt.thy one up to be loaded before Nominal2_Atoms
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 960 apply(simp add: supports_def)
9eab1dfc14d2
moved lemmas from FSet.thy to do with atom to Nominal2_Base, and to do with 'a::at set to Nominal2_Atoms; moved Nominal2_Eqvt.thy one up to be loaded before Nominal2_Atoms
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 961 apply(simp add: swap_set_not_in)
9eab1dfc14d2
moved lemmas from FSet.thy to do with atom to Nominal2_Base, and to do with 'a::at set to Nominal2_Atoms; moved Nominal2_Eqvt.thy one up to be loaded before Nominal2_Atoms
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 962 apply(rule assms)
9eab1dfc14d2
moved lemmas from FSet.thy to do with atom to Nominal2_Base, and to do with 'a::at set to Nominal2_Atoms; moved Nominal2_Eqvt.thy one up to be loaded before Nominal2_Atoms
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 963 apply(simp add: swap_set_in)
9eab1dfc14d2
moved lemmas from FSet.thy to do with atom to Nominal2_Base, and to do with 'a::at set to Nominal2_Atoms; moved Nominal2_Eqvt.thy one up to be loaded before Nominal2_Atoms
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 964 done
9eab1dfc14d2
moved lemmas from FSet.thy to do with atom to Nominal2_Base, and to do with 'a::at set to Nominal2_Atoms; moved Nominal2_Eqvt.thy one up to be loaded before Nominal2_Atoms
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 965
1062
+ − 966 section {* Type @{typ perm} is finitely-supported. *}
+ − 967
+ − 968 lemma perm_swap_eq:
+ − 969 shows "(a \<rightleftharpoons> b) \<bullet> p = p \<longleftrightarrow> (p \<bullet> (a \<rightleftharpoons> b)) = (a \<rightleftharpoons> b)"
+ − 970 unfolding permute_perm_def
+ − 971 by (metis add_diff_cancel minus_perm_def)
+ − 972
+ − 973 lemma supports_perm:
+ − 974 shows "{a. p \<bullet> a \<noteq> a} supports p"
+ − 975 unfolding supports_def
1879
+ − 976 unfolding perm_swap_eq
+ − 977 by (simp add: swap_eqvt)
1062
+ − 978
+ − 979 lemma finite_perm_lemma:
+ − 980 shows "finite {a::atom. p \<bullet> a \<noteq> a}"
+ − 981 using finite_Rep_perm [of p]
+ − 982 unfolding permute_atom_def .
+ − 983
+ − 984 lemma supp_perm:
+ − 985 shows "supp p = {a. p \<bullet> a \<noteq> a}"
+ − 986 apply (rule finite_supp_unique)
+ − 987 apply (rule supports_perm)
+ − 988 apply (rule finite_perm_lemma)
+ − 989 apply (simp add: perm_swap_eq swap_eqvt)
+ − 990 apply (auto simp add: expand_perm_eq swap_atom)
+ − 991 done
+ − 992
+ − 993 lemma fresh_perm:
+ − 994 shows "a \<sharp> p \<longleftrightarrow> p \<bullet> a = a"
1879
+ − 995 unfolding fresh_def
+ − 996 by (simp add: supp_perm)
1062
+ − 997
+ − 998 lemma supp_swap:
+ − 999 shows "supp (a \<rightleftharpoons> b) = (if a = b \<or> sort_of a \<noteq> sort_of b then {} else {a, b})"
+ − 1000 by (auto simp add: supp_perm swap_atom)
+ − 1001
+ − 1002 lemma fresh_zero_perm:
+ − 1003 shows "a \<sharp> (0::perm)"
+ − 1004 unfolding fresh_perm by simp
+ − 1005
+ − 1006 lemma supp_zero_perm:
+ − 1007 shows "supp (0::perm) = {}"
+ − 1008 unfolding supp_perm by simp
+ − 1009
1087
+ − 1010 lemma fresh_plus_perm:
+ − 1011 fixes p q::perm
+ − 1012 assumes "a \<sharp> p" "a \<sharp> q"
+ − 1013 shows "a \<sharp> (p + q)"
+ − 1014 using assms
+ − 1015 unfolding fresh_def
+ − 1016 by (auto simp add: supp_perm)
+ − 1017
1062
+ − 1018 lemma supp_plus_perm:
+ − 1019 fixes p q::perm
+ − 1020 shows "supp (p + q) \<subseteq> supp p \<union> supp q"
+ − 1021 by (auto simp add: supp_perm)
+ − 1022
1087
+ − 1023 lemma fresh_minus_perm:
+ − 1024 fixes p::perm
+ − 1025 shows "a \<sharp> (- p) \<longleftrightarrow> a \<sharp> p"
+ − 1026 unfolding fresh_def
1879
+ − 1027 unfolding supp_perm
+ − 1028 apply(simp)
+ − 1029 apply(metis permute_minus_cancel)
1087
+ − 1030 done
+ − 1031
1062
+ − 1032 lemma supp_minus_perm:
+ − 1033 fixes p::perm
+ − 1034 shows "supp (- p) = supp p"
1087
+ − 1035 unfolding supp_conv_fresh
+ − 1036 by (simp add: fresh_minus_perm)
1062
+ − 1037
+ − 1038 instance perm :: fs
+ − 1039 by default (simp add: supp_perm finite_perm_lemma)
+ − 1040
1305
+ − 1041 lemma plus_perm_eq:
+ − 1042 fixes p q::"perm"
1879
+ − 1043 assumes asm: "supp p \<inter> supp q = {}"
1305
+ − 1044 shows "p + q = q + p"
+ − 1045 unfolding expand_perm_eq
+ − 1046 proof
+ − 1047 fix a::"atom"
+ − 1048 show "(p + q) \<bullet> a = (q + p) \<bullet> a"
+ − 1049 proof -
+ − 1050 { assume "a \<notin> supp p" "a \<notin> supp q"
+ − 1051 then have "(p + q) \<bullet> a = (q + p) \<bullet> a"
+ − 1052 by (simp add: supp_perm)
+ − 1053 }
+ − 1054 moreover
+ − 1055 { assume a: "a \<in> supp p" "a \<notin> supp q"
+ − 1056 then have "p \<bullet> a \<in> supp p" by (simp add: supp_perm)
+ − 1057 then have "p \<bullet> a \<notin> supp q" using asm by auto
+ − 1058 with a have "(p + q) \<bullet> a = (q + p) \<bullet> a"
+ − 1059 by (simp add: supp_perm)
+ − 1060 }
+ − 1061 moreover
+ − 1062 { assume a: "a \<notin> supp p" "a \<in> supp q"
+ − 1063 then have "q \<bullet> a \<in> supp q" by (simp add: supp_perm)
+ − 1064 then have "q \<bullet> a \<notin> supp p" using asm by auto
+ − 1065 with a have "(p + q) \<bullet> a = (q + p) \<bullet> a"
+ − 1066 by (simp add: supp_perm)
+ − 1067 }
+ − 1068 ultimately show "(p + q) \<bullet> a = (q + p) \<bullet> a"
+ − 1069 using asm by blast
+ − 1070 qed
+ − 1071 qed
1062
+ − 1072
+ − 1073 section {* Finite Support instances for other types *}
+ − 1074
2565
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1075
1062
+ − 1076 subsection {* Type @{typ "'a \<times> 'b"} is finitely-supported. *}
+ − 1077
+ − 1078 lemma supp_Pair:
+ − 1079 shows "supp (x, y) = supp x \<union> supp y"
+ − 1080 by (simp add: supp_def Collect_imp_eq Collect_neg_eq)
+ − 1081
+ − 1082 lemma fresh_Pair:
+ − 1083 shows "a \<sharp> (x, y) \<longleftrightarrow> a \<sharp> x \<and> a \<sharp> y"
+ − 1084 by (simp add: fresh_def supp_Pair)
+ − 1085
2470
+ − 1086 lemma supp_Unit:
+ − 1087 shows "supp () = {}"
+ − 1088 by (simp add: supp_def)
+ − 1089
+ − 1090 lemma fresh_Unit:
+ − 1091 shows "a \<sharp> ()"
+ − 1092 by (simp add: fresh_def supp_Unit)
+ − 1093
2378
+ − 1094 instance prod :: (fs, fs) fs
1062
+ − 1095 apply default
+ − 1096 apply (induct_tac x)
+ − 1097 apply (simp add: supp_Pair finite_supp)
+ − 1098 done
+ − 1099
2565
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1100
1062
+ − 1101 subsection {* Type @{typ "'a + 'b"} is finitely supported *}
+ − 1102
+ − 1103 lemma supp_Inl:
+ − 1104 shows "supp (Inl x) = supp x"
+ − 1105 by (simp add: supp_def)
+ − 1106
+ − 1107 lemma supp_Inr:
+ − 1108 shows "supp (Inr x) = supp x"
+ − 1109 by (simp add: supp_def)
+ − 1110
+ − 1111 lemma fresh_Inl:
+ − 1112 shows "a \<sharp> Inl x \<longleftrightarrow> a \<sharp> x"
+ − 1113 by (simp add: fresh_def supp_Inl)
+ − 1114
+ − 1115 lemma fresh_Inr:
+ − 1116 shows "a \<sharp> Inr y \<longleftrightarrow> a \<sharp> y"
+ − 1117 by (simp add: fresh_def supp_Inr)
+ − 1118
2378
+ − 1119 instance sum :: (fs, fs) fs
1062
+ − 1120 apply default
+ − 1121 apply (induct_tac x)
+ − 1122 apply (simp_all add: supp_Inl supp_Inr finite_supp)
+ − 1123 done
+ − 1124
2565
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1125
1062
+ − 1126 subsection {* Type @{typ "'a option"} is finitely supported *}
+ − 1127
+ − 1128 lemma supp_None:
+ − 1129 shows "supp None = {}"
+ − 1130 by (simp add: supp_def)
+ − 1131
+ − 1132 lemma supp_Some:
+ − 1133 shows "supp (Some x) = supp x"
+ − 1134 by (simp add: supp_def)
+ − 1135
+ − 1136 lemma fresh_None:
+ − 1137 shows "a \<sharp> None"
+ − 1138 by (simp add: fresh_def supp_None)
+ − 1139
+ − 1140 lemma fresh_Some:
+ − 1141 shows "a \<sharp> Some x \<longleftrightarrow> a \<sharp> x"
+ − 1142 by (simp add: fresh_def supp_Some)
+ − 1143
+ − 1144 instance option :: (fs) fs
+ − 1145 apply default
+ − 1146 apply (induct_tac x)
+ − 1147 apply (simp_all add: supp_None supp_Some finite_supp)
+ − 1148 done
+ − 1149
2565
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1150
1062
+ − 1151 subsubsection {* Type @{typ "'a list"} is finitely supported *}
+ − 1152
+ − 1153 lemma supp_Nil:
+ − 1154 shows "supp [] = {}"
+ − 1155 by (simp add: supp_def)
+ − 1156
+ − 1157 lemma supp_Cons:
+ − 1158 shows "supp (x # xs) = supp x \<union> supp xs"
+ − 1159 by (simp add: supp_def Collect_imp_eq Collect_neg_eq)
+ − 1160
2591
+ − 1161 lemma supp_append:
+ − 1162 shows "supp (xs @ ys) = supp xs \<union> supp ys"
+ − 1163 by (induct xs) (auto simp add: supp_Nil supp_Cons)
+ − 1164
1062
+ − 1165 lemma fresh_Nil:
+ − 1166 shows "a \<sharp> []"
+ − 1167 by (simp add: fresh_def supp_Nil)
+ − 1168
+ − 1169 lemma fresh_Cons:
+ − 1170 shows "a \<sharp> (x # xs) \<longleftrightarrow> a \<sharp> x \<and> a \<sharp> xs"
+ − 1171 by (simp add: fresh_def supp_Cons)
+ − 1172
2591
+ − 1173 lemma fresh_append:
+ − 1174 shows "a \<sharp> (xs @ ys) \<longleftrightarrow> a \<sharp> xs \<and> a \<sharp> ys"
+ − 1175 by (induct xs) (simp_all add: fresh_Nil fresh_Cons)
+ − 1176
+ − 1177
1062
+ − 1178 instance list :: (fs) fs
+ − 1179 apply default
+ − 1180 apply (induct_tac x)
+ − 1181 apply (simp_all add: supp_Nil supp_Cons finite_supp)
+ − 1182 done
+ − 1183
2588
8f5420681039
completed the strong exhausts rules for Foo2 using general lemmas
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1184 lemma supp_of_atom_list:
8f5420681039
completed the strong exhausts rules for Foo2 using general lemmas
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1185 fixes as::"atom list"
8f5420681039
completed the strong exhausts rules for Foo2 using general lemmas
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1186 shows "supp as = set as"
8f5420681039
completed the strong exhausts rules for Foo2 using general lemmas
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1187 by (induct as)
8f5420681039
completed the strong exhausts rules for Foo2 using general lemmas
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1188 (simp_all add: supp_Nil supp_Cons supp_atom)
8f5420681039
completed the strong exhausts rules for Foo2 using general lemmas
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1189
2466
+ − 1190
2470
+ − 1191 section {* Support and Freshness for Applications *}
1062
+ − 1192
1879
+ − 1193 lemma fresh_conv_MOST:
+ − 1194 shows "a \<sharp> x \<longleftrightarrow> (MOST b. (a \<rightleftharpoons> b) \<bullet> x = x)"
+ − 1195 unfolding fresh_def supp_def
+ − 1196 unfolding MOST_iff_cofinite by simp
+ − 1197
2003
b53e98bfb298
added lemmas establishing the support of finite sets of finitely supported elements
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1198 lemma supp_subset_fresh:
b53e98bfb298
added lemmas establishing the support of finite sets of finitely supported elements
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1199 assumes a: "\<And>a. a \<sharp> x \<Longrightarrow> a \<sharp> y"
b53e98bfb298
added lemmas establishing the support of finite sets of finitely supported elements
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1200 shows "supp y \<subseteq> supp x"
b53e98bfb298
added lemmas establishing the support of finite sets of finitely supported elements
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1201 using a
b53e98bfb298
added lemmas establishing the support of finite sets of finitely supported elements
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1202 unfolding fresh_def
b53e98bfb298
added lemmas establishing the support of finite sets of finitely supported elements
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1203 by blast
b53e98bfb298
added lemmas establishing the support of finite sets of finitely supported elements
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1204
1879
+ − 1205 lemma fresh_fun_app:
+ − 1206 assumes "a \<sharp> f" and "a \<sharp> x"
+ − 1207 shows "a \<sharp> f x"
2003
b53e98bfb298
added lemmas establishing the support of finite sets of finitely supported elements
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1208 using assms
1879
+ − 1209 unfolding fresh_conv_MOST
+ − 1210 unfolding permute_fun_app_eq
+ − 1211 by (elim MOST_rev_mp, simp)
+ − 1212
1062
+ − 1213 lemma supp_fun_app:
+ − 1214 shows "supp (f x) \<subseteq> (supp f) \<union> (supp x)"
1879
+ − 1215 using fresh_fun_app
+ − 1216 unfolding fresh_def
+ − 1217 by auto
+ − 1218
2470
+ − 1219 text {* Support of Equivariant Functions *}
2003
b53e98bfb298
added lemmas establishing the support of finite sets of finitely supported elements
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1220
1941
+ − 1221 lemma supp_fun_eqvt:
2003
b53e98bfb298
added lemmas establishing the support of finite sets of finitely supported elements
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1222 assumes a: "\<And>p. p \<bullet> f = f"
1941
+ − 1223 shows "supp f = {}"
+ − 1224 unfolding supp_def
+ − 1225 using a by simp
+ − 1226
1062
+ − 1227 lemma fresh_fun_eqvt_app:
2003
b53e98bfb298
added lemmas establishing the support of finite sets of finitely supported elements
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1228 assumes a: "\<And>p. p \<bullet> f = f"
1062
+ − 1229 shows "a \<sharp> x \<Longrightarrow> a \<sharp> f x"
+ − 1230 proof -
1941
+ − 1231 from a have "supp f = {}" by (simp add: supp_fun_eqvt)
1879
+ − 1232 then show "a \<sharp> x \<Longrightarrow> a \<sharp> f x"
1062
+ − 1233 unfolding fresh_def
2003
b53e98bfb298
added lemmas establishing the support of finite sets of finitely supported elements
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1234 using supp_fun_app by auto
1062
+ − 1235 qed
+ − 1236
1962
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1237
2466
+ − 1238 section {* Support of Finite Sets of Finitely Supported Elements *}
+ − 1239
+ − 1240 lemma Union_fresh:
+ − 1241 shows "a \<sharp> S \<Longrightarrow> a \<sharp> (\<Union>x \<in> S. supp x)"
+ − 1242 unfolding Union_image_eq[symmetric]
+ − 1243 apply(rule_tac f="\<lambda>S. \<Union> supp ` S" in fresh_fun_eqvt_app)
+ − 1244 apply(simp add: permute_fun_def UNION_def Collect_def Bex_def ex_eqvt mem_def conj_eqvt)
+ − 1245 apply(subst permute_fun_app_eq)
+ − 1246 back
+ − 1247 apply(simp add: supp_eqvt)
+ − 1248 apply(assumption)
+ − 1249 done
+ − 1250
+ − 1251 lemma Union_supports_set:
+ − 1252 shows "(\<Union>x \<in> S. supp x) supports S"
+ − 1253 proof -
+ − 1254 { fix a b
+ − 1255 have "\<forall>x \<in> S. (a \<rightleftharpoons> b) \<bullet> x = x \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> S = S"
+ − 1256 unfolding permute_set_eq by force
+ − 1257 }
+ − 1258 then show "(\<Union>x \<in> S. supp x) supports S"
+ − 1259 unfolding supports_def
+ − 1260 by (simp add: fresh_def[symmetric] swap_fresh_fresh)
+ − 1261 qed
+ − 1262
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+ − 1263 lemma Union_of_finite_supp_sets:
2466
+ − 1264 fixes S::"('a::fs set)"
+ − 1265 assumes fin: "finite S"
+ − 1266 shows "finite (\<Union>x\<in>S. supp x)"
+ − 1267 using fin by (induct) (auto simp add: finite_supp)
+ − 1268
+ − 1269 lemma Union_included_in_supp:
+ − 1270 fixes S::"('a::fs set)"
+ − 1271 assumes fin: "finite S"
+ − 1272 shows "(\<Union>x\<in>S. supp x) \<subseteq> supp S"
+ − 1273 proof -
+ − 1274 have "(\<Union>x\<in>S. supp x) = supp (\<Union>x\<in>S. supp x)"
+ − 1275 by (rule supp_finite_atom_set[symmetric])
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+ − 1276 (rule Union_of_finite_supp_sets[OF fin])
2466
+ − 1277 also have "\<dots> \<subseteq> supp S"
+ − 1278 by (rule supp_subset_fresh)
+ − 1279 (simp add: Union_fresh)
+ − 1280 finally show "(\<Union>x\<in>S. supp x) \<subseteq> supp S" .
+ − 1281 qed
+ − 1282
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+ − 1283 lemma supp_of_finite_sets:
2466
+ − 1284 fixes S::"('a::fs set)"
+ − 1285 assumes fin: "finite S"
+ − 1286 shows "(supp S) = (\<Union>x\<in>S. supp x)"
+ − 1287 apply(rule subset_antisym)
+ − 1288 apply(rule supp_is_subset)
+ − 1289 apply(rule Union_supports_set)
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+ − 1290 apply(rule Union_of_finite_supp_sets[OF fin])
2466
+ − 1291 apply(rule Union_included_in_supp[OF fin])
+ − 1292 done
+ − 1293
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+ − 1294 lemma finite_sets_supp:
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+ − 1295 fixes S::"('a::fs set)"
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+ − 1296 assumes "finite S"
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+ − 1297 shows "finite (supp S)"
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+ − 1298 using assms
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+ − 1299 by (simp only: supp_of_finite_sets Union_of_finite_supp_sets)
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+ − 1300
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+ − 1301 lemma supp_of_finite_union:
2466
+ − 1302 fixes S T::"('a::fs) set"
+ − 1303 assumes fin1: "finite S"
+ − 1304 and fin2: "finite T"
+ − 1305 shows "supp (S \<union> T) = supp S \<union> supp T"
+ − 1306 using fin1 fin2
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+ − 1307 by (simp add: supp_of_finite_sets)
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+ − 1308
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+ − 1309 lemma supp_of_finite_insert:
2466
+ − 1310 fixes S::"('a::fs) set"
+ − 1311 assumes fin: "finite S"
+ − 1312 shows "supp (insert x S) = supp x \<union> supp S"
+ − 1313 using fin
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+ − 1314 by (simp add: supp_of_finite_sets)
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+ − 1315
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+ − 1316 lemma fresh_finite_insert:
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+ − 1317 fixes S::"('a::fs) set"
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+ − 1318 assumes fin: "finite S"
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+ − 1319 shows "a \<sharp> (insert x S) \<longleftrightarrow> a \<sharp> x \<and> a \<sharp> S"
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+ − 1320 using fin unfolding fresh_def
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+ − 1321 by (simp add: supp_of_finite_insert)
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+ − 1322
2591
+ − 1323 lemma supp_set_empty:
+ − 1324 shows "supp {} = {}"
+ − 1325 unfolding supp_def
+ − 1326 by (simp add: empty_eqvt)
+ − 1327
+ − 1328 lemma fresh_set_empty:
+ − 1329 shows "a \<sharp> {}"
+ − 1330 by (simp add: fresh_def supp_set_empty)
+ − 1331
+ − 1332 lemma supp_set:
+ − 1333 fixes xs :: "('a::fs) list"
+ − 1334 shows "supp (set xs) = supp xs"
+ − 1335 apply(induct xs)
+ − 1336 apply(simp add: supp_set_empty supp_Nil)
+ − 1337 apply(simp add: supp_Cons supp_of_finite_insert)
+ − 1338 done
+ − 1339
+ − 1340 lemma fresh_set:
+ − 1341 fixes xs :: "('a::fs) list"
+ − 1342 shows "a \<sharp> (set xs) \<longleftrightarrow> a \<sharp> xs"
+ − 1343 unfolding fresh_def
+ − 1344 by (simp add: supp_set)
+ − 1345
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+ − 1346
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+ − 1347 subsection {* Type @{typ "'a fset"} is finitely supported *}
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+ − 1348
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+ − 1349 lemma fset_eqvt:
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+ − 1350 shows "p \<bullet> (fset S) = fset (p \<bullet> S)"
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+ − 1351 by (lifting set_eqvt)
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+ − 1352
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+ − 1353 lemma supp_fset [simp]:
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+ − 1354 shows "supp (fset S) = supp S"
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+ − 1355 unfolding supp_def
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+ − 1356 by (simp add: fset_eqvt fset_cong)
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+ − 1357
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+ − 1358 lemma supp_empty_fset [simp]:
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+ − 1359 shows "supp {||} = {}"
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+ − 1360 unfolding supp_def
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+ − 1361 by simp
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+ − 1362
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+ − 1363 lemma supp_insert_fset [simp]:
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+ − 1364 fixes x::"'a::fs"
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+ − 1365 and S::"'a fset"
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+ − 1366 shows "supp (insert_fset x S) = supp x \<union> supp S"
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+ − 1367 apply(subst supp_fset[symmetric])
2587
+ − 1368 apply(simp add: supp_of_finite_insert)
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+ − 1369 done
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+ − 1370
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+ − 1371 lemma fset_finite_supp:
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+ − 1372 fixes S::"('a::fs) fset"
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+ − 1373 shows "finite (supp S)"
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+ − 1374 by (induct S) (simp_all add: finite_supp)
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+ − 1375
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+ − 1376
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+ − 1377 instance fset :: (fs) fs
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+ − 1378 apply (default)
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+ − 1379 apply (rule fset_finite_supp)
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+ − 1380 done
2466
+ − 1381
+ − 1382
2470
+ − 1383 section {* Fresh-Star *}
+ − 1384
+ − 1385
+ − 1386 text {* The fresh-star generalisation of fresh is used in strong
+ − 1387 induction principles. *}
+ − 1388
+ − 1389 definition
+ − 1390 fresh_star :: "atom set \<Rightarrow> 'a::pt \<Rightarrow> bool" ("_ \<sharp>* _" [80,80] 80)
+ − 1391 where
+ − 1392 "as \<sharp>* x \<equiv> \<forall>a \<in> as. a \<sharp> x"
+ − 1393
2507
+ − 1394 lemma fresh_star_supp_conv:
+ − 1395 shows "supp x \<sharp>* y \<Longrightarrow> supp y \<sharp>* x"
+ − 1396 by (auto simp add: fresh_star_def fresh_def)
+ − 1397
2591
+ − 1398 lemma fresh_star_Pair:
2470
+ − 1399 shows "as \<sharp>* (x, y) = (as \<sharp>* x \<and> as \<sharp>* y)"
+ − 1400 by (auto simp add: fresh_star_def fresh_Pair)
+ − 1401
2591
+ − 1402 lemma fresh_star_list:
+ − 1403 shows "as \<sharp>* (xs @ ys) \<longleftrightarrow> as \<sharp>* xs \<and> as \<sharp>* ys"
+ − 1404 and "as \<sharp>* (x # xs) \<longleftrightarrow> as \<sharp>* x \<and> as \<sharp>* xs"
+ − 1405 and "as \<sharp>* []"
+ − 1406 by (auto simp add: fresh_star_def fresh_Nil fresh_Cons fresh_append)
+ − 1407
+ − 1408 lemma fresh_star_set:
+ − 1409 fixes xs::"('a::fs) list"
+ − 1410 shows "as \<sharp>* set xs \<longleftrightarrow> as \<sharp>* xs"
+ − 1411 unfolding fresh_star_def
+ − 1412 by (simp add: fresh_set)
+ − 1413
+ − 1414 lemma fresh_star_Un:
2470
+ − 1415 shows "(as \<union> bs) \<sharp>* x = (as \<sharp>* x \<and> bs \<sharp>* x)"
+ − 1416 by (auto simp add: fresh_star_def)
+ − 1417
+ − 1418 lemma fresh_star_insert:
+ − 1419 shows "(insert a as) \<sharp>* x = (a \<sharp> x \<and> as \<sharp>* x)"
+ − 1420 by (auto simp add: fresh_star_def)
+ − 1421
+ − 1422 lemma fresh_star_Un_elim:
+ − 1423 "((as \<union> bs) \<sharp>* x \<Longrightarrow> PROP C) \<equiv> (as \<sharp>* x \<Longrightarrow> bs \<sharp>* x \<Longrightarrow> PROP C)"
+ − 1424 unfolding fresh_star_def
+ − 1425 apply(rule)
+ − 1426 apply(erule meta_mp)
+ − 1427 apply(auto)
+ − 1428 done
+ − 1429
+ − 1430 lemma fresh_star_insert_elim:
+ − 1431 "(insert a as \<sharp>* x \<Longrightarrow> PROP C) \<equiv> (a \<sharp> x \<Longrightarrow> as \<sharp>* x \<Longrightarrow> PROP C)"
+ − 1432 unfolding fresh_star_def
+ − 1433 by rule (simp_all add: fresh_star_def)
+ − 1434
+ − 1435 lemma fresh_star_empty_elim:
+ − 1436 "({} \<sharp>* x \<Longrightarrow> PROP C) \<equiv> PROP C"
+ − 1437 by (simp add: fresh_star_def)
+ − 1438
+ − 1439 lemma fresh_star_unit_elim:
+ − 1440 shows "(a \<sharp>* () \<Longrightarrow> PROP C) \<equiv> PROP C"
+ − 1441 by (simp add: fresh_star_def fresh_Unit)
+ − 1442
2591
+ − 1443 lemma fresh_star_Pair_elim:
2470
+ − 1444 shows "(a \<sharp>* (x, y) \<Longrightarrow> PROP C) \<equiv> (a \<sharp>* x \<Longrightarrow> a \<sharp>* y \<Longrightarrow> PROP C)"
2591
+ − 1445 by (rule, simp_all add: fresh_star_Pair)
2470
+ − 1446
+ − 1447 lemma fresh_star_zero:
+ − 1448 shows "as \<sharp>* (0::perm)"
+ − 1449 unfolding fresh_star_def
+ − 1450 by (simp add: fresh_zero_perm)
+ − 1451
+ − 1452 lemma fresh_star_plus:
+ − 1453 fixes p q::perm
+ − 1454 shows "\<lbrakk>a \<sharp>* p; a \<sharp>* q\<rbrakk> \<Longrightarrow> a \<sharp>* (p + q)"
+ − 1455 unfolding fresh_star_def
+ − 1456 by (simp add: fresh_plus_perm)
+ − 1457
+ − 1458 lemma fresh_star_permute_iff:
+ − 1459 shows "(p \<bullet> a) \<sharp>* (p \<bullet> x) \<longleftrightarrow> a \<sharp>* x"
+ − 1460 unfolding fresh_star_def
+ − 1461 by (metis mem_permute_iff permute_minus_cancel(1) fresh_permute_iff)
+ − 1462
+ − 1463 lemma fresh_star_eqvt:
+ − 1464 shows "(p \<bullet> (as \<sharp>* x)) = (p \<bullet> as) \<sharp>* (p \<bullet> x)"
+ − 1465 unfolding fresh_star_def
+ − 1466 unfolding Ball_def
+ − 1467 apply(simp add: all_eqvt)
+ − 1468 apply(subst permute_fun_def)
+ − 1469 apply(simp add: imp_eqvt fresh_eqvt mem_eqvt)
+ − 1470 done
+ − 1471
2591
+ − 1472 lemma at_fresh_star_inter:
+ − 1473 assumes a: "(p \<bullet> bs) \<sharp>* bs"
+ − 1474 and b: "finite bs"
+ − 1475 shows "p \<bullet> bs \<inter> bs = {}"
+ − 1476 using a b
+ − 1477 unfolding fresh_star_def
+ − 1478 unfolding fresh_def
+ − 1479 by (auto simp add: supp_finite_atom_set)
+ − 1480
2470
+ − 1481
+ − 1482 section {* Induction principle for permutations *}
+ − 1483
+ − 1484
+ − 1485 lemma perm_struct_induct[consumes 1, case_names zero swap]:
+ − 1486 assumes S: "supp p \<subseteq> S"
+ − 1487 and zero: "P 0"
+ − 1488 and swap: "\<And>p a b. \<lbrakk>P p; supp p \<subseteq> S; a \<in> S; b \<in> S; a \<noteq> b; sort_of a = sort_of b\<rbrakk> \<Longrightarrow> P ((a \<rightleftharpoons> b) + p)"
+ − 1489 shows "P p"
+ − 1490 proof -
+ − 1491 have "finite (supp p)" by (simp add: finite_supp)
+ − 1492 then show "P p" using S
+ − 1493 proof(induct A\<equiv>"supp p" arbitrary: p rule: finite_psubset_induct)
+ − 1494 case (psubset p)
+ − 1495 then have ih: "\<And>q. supp q \<subset> supp p \<Longrightarrow> P q" by auto
+ − 1496 have as: "supp p \<subseteq> S" by fact
+ − 1497 { assume "supp p = {}"
+ − 1498 then have "p = 0" by (simp add: supp_perm expand_perm_eq)
+ − 1499 then have "P p" using zero by simp
+ − 1500 }
+ − 1501 moreover
+ − 1502 { assume "supp p \<noteq> {}"
+ − 1503 then obtain a where a0: "a \<in> supp p" by blast
+ − 1504 then have a1: "p \<bullet> a \<in> S" "a \<in> S" "sort_of (p \<bullet> a) = sort_of a" "p \<bullet> a \<noteq> a"
+ − 1505 using as by (auto simp add: supp_atom supp_perm swap_atom)
+ − 1506 let ?q = "(p \<bullet> a \<rightleftharpoons> a) + p"
+ − 1507 have a2: "supp ?q \<subseteq> supp p" unfolding supp_perm by (auto simp add: swap_atom)
+ − 1508 moreover
+ − 1509 have "a \<notin> supp ?q" by (simp add: supp_perm)
+ − 1510 then have "supp ?q \<noteq> supp p" using a0 by auto
+ − 1511 ultimately have "supp ?q \<subset> supp p" using a2 by auto
+ − 1512 then have "P ?q" using ih by simp
+ − 1513 moreover
+ − 1514 have "supp ?q \<subseteq> S" using as a2 by simp
+ − 1515 ultimately have "P ((p \<bullet> a \<rightleftharpoons> a) + ?q)" using as a1 swap by simp
+ − 1516 moreover
+ − 1517 have "p = (p \<bullet> a \<rightleftharpoons> a) + ?q" by (simp add: expand_perm_eq)
+ − 1518 ultimately have "P p" by simp
+ − 1519 }
+ − 1520 ultimately show "P p" by blast
+ − 1521 qed
+ − 1522 qed
+ − 1523
+ − 1524 lemma perm_simple_struct_induct[case_names zero swap]:
+ − 1525 assumes zero: "P 0"
+ − 1526 and swap: "\<And>p a b. \<lbrakk>P p; a \<noteq> b; sort_of a = sort_of b\<rbrakk> \<Longrightarrow> P ((a \<rightleftharpoons> b) + p)"
+ − 1527 shows "P p"
+ − 1528 by (rule_tac S="supp p" in perm_struct_induct)
+ − 1529 (auto intro: zero swap)
+ − 1530
+ − 1531 lemma perm_subset_induct[consumes 1, case_names zero swap plus]:
+ − 1532 assumes S: "supp p \<subseteq> S"
+ − 1533 assumes zero: "P 0"
+ − 1534 assumes swap: "\<And>a b. \<lbrakk>sort_of a = sort_of b; a \<noteq> b; a \<in> S; b \<in> S\<rbrakk> \<Longrightarrow> P (a \<rightleftharpoons> b)"
+ − 1535 assumes plus: "\<And>p1 p2. \<lbrakk>P p1; P p2; supp p1 \<subseteq> S; supp p2 \<subseteq> S\<rbrakk> \<Longrightarrow> P (p1 + p2)"
+ − 1536 shows "P p"
+ − 1537 using S
+ − 1538 by (induct p rule: perm_struct_induct)
+ − 1539 (auto intro: zero plus swap simp add: supp_swap)
+ − 1540
+ − 1541 lemma supp_perm_eq:
+ − 1542 assumes "(supp x) \<sharp>* p"
+ − 1543 shows "p \<bullet> x = x"
+ − 1544 proof -
+ − 1545 from assms have "supp p \<subseteq> {a. a \<sharp> x}"
+ − 1546 unfolding supp_perm fresh_star_def fresh_def by auto
+ − 1547 then show "p \<bullet> x = x"
+ − 1548 proof (induct p rule: perm_struct_induct)
+ − 1549 case zero
+ − 1550 show "0 \<bullet> x = x" by simp
+ − 1551 next
+ − 1552 case (swap p a b)
+ − 1553 then have "a \<sharp> x" "b \<sharp> x" "p \<bullet> x = x" by simp_all
+ − 1554 then show "((a \<rightleftharpoons> b) + p) \<bullet> x = x" by (simp add: swap_fresh_fresh)
+ − 1555 qed
+ − 1556 qed
+ − 1557
+ − 1558 lemma supp_perm_eq_test:
+ − 1559 assumes "(supp x) \<sharp>* p"
+ − 1560 shows "p \<bullet> x = x"
+ − 1561 proof -
+ − 1562 from assms have "supp p \<subseteq> {a. a \<sharp> x}"
+ − 1563 unfolding supp_perm fresh_star_def fresh_def by auto
+ − 1564 then show "p \<bullet> x = x"
+ − 1565 proof (induct p rule: perm_subset_induct)
+ − 1566 case zero
+ − 1567 show "0 \<bullet> x = x" by simp
+ − 1568 next
+ − 1569 case (swap a b)
+ − 1570 then have "a \<sharp> x" "b \<sharp> x" by simp_all
+ − 1571 then show "(a \<rightleftharpoons> b) \<bullet> x = x" by (simp add: swap_fresh_fresh)
+ − 1572 next
+ − 1573 case (plus p1 p2)
+ − 1574 have "p1 \<bullet> x = x" "p2 \<bullet> x = x" by fact+
+ − 1575 then show "(p1 + p2) \<bullet> x = x" by simp
+ − 1576 qed
+ − 1577 qed
+ − 1578
2591
+ − 1579 lemma perm_supp_eq:
+ − 1580 assumes a: "(supp p) \<sharp>* x"
+ − 1581 shows "p \<bullet> x = x"
+ − 1582 by (rule supp_perm_eq)
+ − 1583 (simp add: fresh_star_supp_conv a)
+ − 1584
2470
+ − 1585
+ − 1586 section {* Avoiding of atom sets *}
+ − 1587
+ − 1588 text {*
+ − 1589 For every set of atoms, there is another set of atoms
+ − 1590 avoiding a finitely supported c and there is a permutation
+ − 1591 which 'translates' between both sets.
+ − 1592 *}
+ − 1593
+ − 1594 lemma at_set_avoiding_aux:
+ − 1595 fixes Xs::"atom set"
+ − 1596 and As::"atom set"
+ − 1597 assumes b: "Xs \<subseteq> As"
+ − 1598 and c: "finite As"
+ − 1599 shows "\<exists>p. (p \<bullet> Xs) \<inter> As = {} \<and> (supp p) \<subseteq> (Xs \<union> (p \<bullet> Xs))"
+ − 1600 proof -
+ − 1601 from b c have "finite Xs" by (rule finite_subset)
+ − 1602 then show ?thesis using b
+ − 1603 proof (induct rule: finite_subset_induct)
+ − 1604 case empty
+ − 1605 have "0 \<bullet> {} \<inter> As = {}" by simp
+ − 1606 moreover
+ − 1607 have "supp (0::perm) \<subseteq> {} \<union> 0 \<bullet> {}" by (simp add: supp_zero_perm)
+ − 1608 ultimately show ?case by blast
+ − 1609 next
+ − 1610 case (insert x Xs)
+ − 1611 then obtain p where
+ − 1612 p1: "(p \<bullet> Xs) \<inter> As = {}" and
+ − 1613 p2: "supp p \<subseteq> (Xs \<union> (p \<bullet> Xs))" by blast
+ − 1614 from `x \<in> As` p1 have "x \<notin> p \<bullet> Xs" by fast
+ − 1615 with `x \<notin> Xs` p2 have "x \<notin> supp p" by fast
+ − 1616 hence px: "p \<bullet> x = x" unfolding supp_perm by simp
+ − 1617 have "finite (As \<union> p \<bullet> Xs)"
+ − 1618 using `finite As` `finite Xs`
+ − 1619 by (simp add: permute_set_eq_image)
+ − 1620 then obtain y where "y \<notin> (As \<union> p \<bullet> Xs)" "sort_of y = sort_of x"
+ − 1621 by (rule obtain_atom)
+ − 1622 hence y: "y \<notin> As" "y \<notin> p \<bullet> Xs" "sort_of y = sort_of x"
+ − 1623 by simp_all
+ − 1624 let ?q = "(x \<rightleftharpoons> y) + p"
+ − 1625 have q: "?q \<bullet> insert x Xs = insert y (p \<bullet> Xs)"
+ − 1626 unfolding insert_eqvt
+ − 1627 using `p \<bullet> x = x` `sort_of y = sort_of x`
+ − 1628 using `x \<notin> p \<bullet> Xs` `y \<notin> p \<bullet> Xs`
+ − 1629 by (simp add: swap_atom swap_set_not_in)
+ − 1630 have "?q \<bullet> insert x Xs \<inter> As = {}"
+ − 1631 using `y \<notin> As` `p \<bullet> Xs \<inter> As = {}`
+ − 1632 unfolding q by simp
+ − 1633 moreover
+ − 1634 have "supp ?q \<subseteq> insert x Xs \<union> ?q \<bullet> insert x Xs"
+ − 1635 using p2 unfolding q
+ − 1636 by (intro subset_trans [OF supp_plus_perm])
+ − 1637 (auto simp add: supp_swap)
+ − 1638 ultimately show ?case by blast
+ − 1639 qed
+ − 1640 qed
+ − 1641
+ − 1642 lemma at_set_avoiding:
+ − 1643 assumes a: "finite Xs"
+ − 1644 and b: "finite (supp c)"
+ − 1645 obtains p::"perm" where "(p \<bullet> Xs)\<sharp>*c" and "(supp p) \<subseteq> (Xs \<union> (p \<bullet> Xs))"
+ − 1646 using a b at_set_avoiding_aux [where Xs="Xs" and As="Xs \<union> supp c"]
+ − 1647 unfolding fresh_star_def fresh_def by blast
+ − 1648
2589
+ − 1649 lemma at_set_avoiding1:
+ − 1650 assumes "finite xs"
+ − 1651 and "finite (supp c)"
+ − 1652 shows "\<exists>p. (p \<bullet> xs) \<sharp>* c"
+ − 1653 using assms
+ − 1654 apply(erule_tac c="c" in at_set_avoiding)
+ − 1655 apply(auto)
+ − 1656 done
+ − 1657
2470
+ − 1658 lemma at_set_avoiding2:
+ − 1659 assumes "finite xs"
+ − 1660 and "finite (supp c)" "finite (supp x)"
+ − 1661 and "xs \<sharp>* x"
+ − 1662 shows "\<exists>p. (p \<bullet> xs) \<sharp>* c \<and> supp x \<sharp>* p"
+ − 1663 using assms
+ − 1664 apply(erule_tac c="(c, x)" in at_set_avoiding)
+ − 1665 apply(simp add: supp_Pair)
+ − 1666 apply(rule_tac x="p" in exI)
2591
+ − 1667 apply(simp add: fresh_star_Pair)
2507
+ − 1668 apply(rule fresh_star_supp_conv)
+ − 1669 apply(auto simp add: fresh_star_def)
2470
+ − 1670 done
+ − 1671
2573
+ − 1672 lemma at_set_avoiding3:
+ − 1673 assumes "finite xs"
+ − 1674 and "finite (supp c)" "finite (supp x)"
+ − 1675 and "xs \<sharp>* x"
2586
+ − 1676 shows "\<exists>p. (p \<bullet> xs) \<sharp>* c \<and> supp x \<sharp>* p \<and> supp p \<subseteq> xs \<union> (p \<bullet> xs)"
+ − 1677 using assms
+ − 1678 apply(erule_tac c="(c, x)" in at_set_avoiding)
+ − 1679 apply(simp add: supp_Pair)
+ − 1680 apply(rule_tac x="p" in exI)
2591
+ − 1681 apply(simp add: fresh_star_Pair)
2586
+ − 1682 apply(rule fresh_star_supp_conv)
+ − 1683 apply(auto simp add: fresh_star_def)
+ − 1684 done
2573
+ − 1685
+ − 1686
2470
+ − 1687 lemma at_set_avoiding2_atom:
+ − 1688 assumes "finite (supp c)" "finite (supp x)"
+ − 1689 and b: "a \<sharp> x"
+ − 1690 shows "\<exists>p. (p \<bullet> a) \<sharp> c \<and> supp x \<sharp>* p"
+ − 1691 proof -
+ − 1692 have a: "{a} \<sharp>* x" unfolding fresh_star_def by (simp add: b)
+ − 1693 obtain p where p1: "(p \<bullet> {a}) \<sharp>* c" and p2: "supp x \<sharp>* p"
+ − 1694 using at_set_avoiding2[of "{a}" "c" "x"] assms a by blast
+ − 1695 have c: "(p \<bullet> a) \<sharp> c" using p1
+ − 1696 unfolding fresh_star_def Ball_def
+ − 1697 by(erule_tac x="p \<bullet> a" in allE) (simp add: permute_set_eq)
+ − 1698 hence "p \<bullet> a \<sharp> c \<and> supp x \<sharp>* p" using p2 by blast
+ − 1699 then show "\<exists>p. (p \<bullet> a) \<sharp> c \<and> supp x \<sharp>* p" by blast
+ − 1700 qed
+ − 1701
+ − 1702
+ − 1703 section {* Concrete Atoms Types *}
1962
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1704
1972
+ − 1705 text {*
+ − 1706 Class @{text at_base} allows types containing multiple sorts of atoms.
+ − 1707 Class @{text at} only allows types with a single sort.
+ − 1708 *}
+ − 1709
1962
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1710 class at_base = pt +
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1711 fixes atom :: "'a \<Rightarrow> atom"
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1712 assumes atom_eq_iff [simp]: "atom a = atom b \<longleftrightarrow> a = b"
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1713 assumes atom_eqvt: "p \<bullet> (atom a) = atom (p \<bullet> a)"
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1714
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1715 class at = at_base +
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1716 assumes sort_of_atom_eq [simp]: "sort_of (atom a) = sort_of (atom b)"
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1717
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1718 lemma supp_at_base:
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1719 fixes a::"'a::at_base"
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1720 shows "supp a = {atom a}"
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1721 by (simp add: supp_atom [symmetric] supp_def atom_eqvt)
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1722
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1723 lemma fresh_at_base:
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1724 shows "a \<sharp> b \<longleftrightarrow> a \<noteq> atom b"
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1725 unfolding fresh_def by (simp add: supp_at_base)
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1726
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1727 instance at_base < fs
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1728 proof qed (simp add: supp_at_base)
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1729
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1730 lemma at_base_infinite [simp]:
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1731 shows "infinite (UNIV :: 'a::at_base set)" (is "infinite ?U")
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1732 proof
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1733 obtain a :: 'a where "True" by auto
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1734 assume "finite ?U"
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1735 hence "finite (atom ` ?U)"
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1736 by (rule finite_imageI)
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1737 then obtain b where b: "b \<notin> atom ` ?U" "sort_of b = sort_of (atom a)"
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1738 by (rule obtain_atom)
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1739 from b(2) have "b = atom ((atom a \<rightleftharpoons> b) \<bullet> a)"
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1740 unfolding atom_eqvt [symmetric]
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1741 by (simp add: swap_atom)
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1742 hence "b \<in> atom ` ?U" by simp
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1743 with b(1) show "False" by simp
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1744 qed
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1745
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1746 lemma swap_at_base_simps [simp]:
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1747 fixes x y::"'a::at_base"
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1748 shows "sort_of (atom x) = sort_of (atom y) \<Longrightarrow> (atom x \<rightleftharpoons> atom y) \<bullet> x = y"
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1749 and "sort_of (atom x) = sort_of (atom y) \<Longrightarrow> (atom x \<rightleftharpoons> atom y) \<bullet> y = x"
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1750 and "atom x \<noteq> a \<Longrightarrow> atom x \<noteq> b \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> x = x"
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1751 unfolding atom_eq_iff [symmetric]
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1752 unfolding atom_eqvt [symmetric]
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1753 by simp_all
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1754
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1755 lemma obtain_at_base:
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1756 assumes X: "finite X"
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1757 obtains a::"'a::at_base" where "atom a \<notin> X"
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1758 proof -
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1759 have "inj (atom :: 'a \<Rightarrow> atom)"
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1760 by (simp add: inj_on_def)
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1761 with X have "finite (atom -` X :: 'a set)"
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1762 by (rule finite_vimageI)
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1763 with at_base_infinite have "atom -` X \<noteq> (UNIV :: 'a set)"
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1764 by auto
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1765 then obtain a :: 'a where "atom a \<notin> X"
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1766 by auto
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1767 thus ?thesis ..
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1768 qed
84a13d1e2511
moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1769
1973
+ − 1770 lemma supp_finite_set_at_base:
1971
8daf6ff5e11a
simpliied and moved the remaining lemmas about the atom-function to Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1771 assumes a: "finite S"
8daf6ff5e11a
simpliied and moved the remaining lemmas about the atom-function to Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1772 shows "supp S = atom ` S"
2565
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1773 apply(simp add: supp_of_finite_sets[OF a])
2466
+ − 1774 apply(simp add: supp_at_base)
+ − 1775 apply(auto)
+ − 1776 done
1971
8daf6ff5e11a
simpliied and moved the remaining lemmas about the atom-function to Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1777
2467
+ − 1778 section {* Infrastructure for concrete atom types *}
1971
8daf6ff5e11a
simpliied and moved the remaining lemmas about the atom-function to Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1779
2467
+ − 1780 section {* A swapping operation for concrete atoms *}
+ − 1781
+ − 1782 definition
+ − 1783 flip :: "'a::at_base \<Rightarrow> 'a \<Rightarrow> perm" ("'(_ \<leftrightarrow> _')")
+ − 1784 where
+ − 1785 "(a \<leftrightarrow> b) = (atom a \<rightleftharpoons> atom b)"
+ − 1786
+ − 1787 lemma flip_self [simp]: "(a \<leftrightarrow> a) = 0"
+ − 1788 unfolding flip_def by (rule swap_self)
+ − 1789
+ − 1790 lemma flip_commute: "(a \<leftrightarrow> b) = (b \<leftrightarrow> a)"
+ − 1791 unfolding flip_def by (rule swap_commute)
+ − 1792
+ − 1793 lemma minus_flip [simp]: "- (a \<leftrightarrow> b) = (a \<leftrightarrow> b)"
+ − 1794 unfolding flip_def by (rule minus_swap)
+ − 1795
+ − 1796 lemma add_flip_cancel: "(a \<leftrightarrow> b) + (a \<leftrightarrow> b) = 0"
+ − 1797 unfolding flip_def by (rule swap_cancel)
+ − 1798
+ − 1799 lemma permute_flip_cancel [simp]: "(a \<leftrightarrow> b) \<bullet> (a \<leftrightarrow> b) \<bullet> x = x"
+ − 1800 unfolding permute_plus [symmetric] add_flip_cancel by simp
+ − 1801
+ − 1802 lemma permute_flip_cancel2 [simp]: "(a \<leftrightarrow> b) \<bullet> (b \<leftrightarrow> a) \<bullet> x = x"
+ − 1803 by (simp add: flip_commute)
+ − 1804
+ − 1805 lemma flip_eqvt:
+ − 1806 fixes a b c::"'a::at_base"
+ − 1807 shows "p \<bullet> (a \<leftrightarrow> b) = (p \<bullet> a \<leftrightarrow> p \<bullet> b)"
+ − 1808 unfolding flip_def
+ − 1809 by (simp add: swap_eqvt atom_eqvt)
+ − 1810
+ − 1811 lemma flip_at_base_simps [simp]:
+ − 1812 shows "sort_of (atom a) = sort_of (atom b) \<Longrightarrow> (a \<leftrightarrow> b) \<bullet> a = b"
+ − 1813 and "sort_of (atom a) = sort_of (atom b) \<Longrightarrow> (a \<leftrightarrow> b) \<bullet> b = a"
+ − 1814 and "\<lbrakk>a \<noteq> c; b \<noteq> c\<rbrakk> \<Longrightarrow> (a \<leftrightarrow> b) \<bullet> c = c"
+ − 1815 and "sort_of (atom a) \<noteq> sort_of (atom b) \<Longrightarrow> (a \<leftrightarrow> b) \<bullet> x = x"
+ − 1816 unfolding flip_def
+ − 1817 unfolding atom_eq_iff [symmetric]
+ − 1818 unfolding atom_eqvt [symmetric]
+ − 1819 by simp_all
+ − 1820
+ − 1821 text {* the following two lemmas do not hold for at_base,
+ − 1822 only for single sort atoms from at *}
+ − 1823
+ − 1824 lemma permute_flip_at:
+ − 1825 fixes a b c::"'a::at"
+ − 1826 shows "(a \<leftrightarrow> b) \<bullet> c = (if c = a then b else if c = b then a else c)"
+ − 1827 unfolding flip_def
+ − 1828 apply (rule atom_eq_iff [THEN iffD1])
+ − 1829 apply (subst atom_eqvt [symmetric])
+ − 1830 apply (simp add: swap_atom)
+ − 1831 done
+ − 1832
+ − 1833 lemma flip_at_simps [simp]:
+ − 1834 fixes a b::"'a::at"
+ − 1835 shows "(a \<leftrightarrow> b) \<bullet> a = b"
+ − 1836 and "(a \<leftrightarrow> b) \<bullet> b = a"
+ − 1837 unfolding permute_flip_at by simp_all
+ − 1838
+ − 1839 lemma flip_fresh_fresh:
+ − 1840 fixes a b::"'a::at_base"
+ − 1841 assumes "atom a \<sharp> x" "atom b \<sharp> x"
+ − 1842 shows "(a \<leftrightarrow> b) \<bullet> x = x"
+ − 1843 using assms
+ − 1844 by (simp add: flip_def swap_fresh_fresh)
+ − 1845
+ − 1846 subsection {* Syntax for coercing at-elements to the atom-type *}
+ − 1847
+ − 1848 syntax
+ − 1849 "_atom_constrain" :: "logic \<Rightarrow> type \<Rightarrow> logic" ("_:::_" [4, 0] 3)
+ − 1850
+ − 1851 translations
+ − 1852 "_atom_constrain a t" => "CONST atom (_constrain a t)"
+ − 1853
+ − 1854
+ − 1855 subsection {* A lemma for proving instances of class @{text at}. *}
+ − 1856
+ − 1857 setup {* Sign.add_const_constraint (@{const_name "permute"}, NONE) *}
+ − 1858 setup {* Sign.add_const_constraint (@{const_name "atom"}, NONE) *}
+ − 1859
+ − 1860 text {*
+ − 1861 New atom types are defined as subtypes of @{typ atom}.
+ − 1862 *}
+ − 1863
+ − 1864 lemma exists_eq_simple_sort:
+ − 1865 shows "\<exists>a. a \<in> {a. sort_of a = s}"
+ − 1866 by (rule_tac x="Atom s 0" in exI, simp)
+ − 1867
+ − 1868 lemma exists_eq_sort:
+ − 1869 shows "\<exists>a. a \<in> {a. sort_of a \<in> range sort_fun}"
+ − 1870 by (rule_tac x="Atom (sort_fun x) y" in exI, simp)
+ − 1871
+ − 1872 lemma at_base_class:
+ − 1873 fixes sort_fun :: "'b \<Rightarrow>atom_sort"
+ − 1874 fixes Rep :: "'a \<Rightarrow> atom" and Abs :: "atom \<Rightarrow> 'a"
+ − 1875 assumes type: "type_definition Rep Abs {a. sort_of a \<in> range sort_fun}"
+ − 1876 assumes atom_def: "\<And>a. atom a = Rep a"
+ − 1877 assumes permute_def: "\<And>p a. p \<bullet> a = Abs (p \<bullet> Rep a)"
+ − 1878 shows "OFCLASS('a, at_base_class)"
+ − 1879 proof
+ − 1880 interpret type_definition Rep Abs "{a. sort_of a \<in> range sort_fun}" by (rule type)
+ − 1881 have sort_of_Rep: "\<And>a. sort_of (Rep a) \<in> range sort_fun" using Rep by simp
+ − 1882 fix a b :: 'a and p p1 p2 :: perm
+ − 1883 show "0 \<bullet> a = a"
+ − 1884 unfolding permute_def by (simp add: Rep_inverse)
+ − 1885 show "(p1 + p2) \<bullet> a = p1 \<bullet> p2 \<bullet> a"
+ − 1886 unfolding permute_def by (simp add: Abs_inverse sort_of_Rep)
+ − 1887 show "atom a = atom b \<longleftrightarrow> a = b"
+ − 1888 unfolding atom_def by (simp add: Rep_inject)
+ − 1889 show "p \<bullet> atom a = atom (p \<bullet> a)"
+ − 1890 unfolding permute_def atom_def by (simp add: Abs_inverse sort_of_Rep)
+ − 1891 qed
+ − 1892
+ − 1893 (*
+ − 1894 lemma at_class:
+ − 1895 fixes s :: atom_sort
+ − 1896 fixes Rep :: "'a \<Rightarrow> atom" and Abs :: "atom \<Rightarrow> 'a"
+ − 1897 assumes type: "type_definition Rep Abs {a. sort_of a \<in> range (\<lambda>x::unit. s)}"
+ − 1898 assumes atom_def: "\<And>a. atom a = Rep a"
+ − 1899 assumes permute_def: "\<And>p a. p \<bullet> a = Abs (p \<bullet> Rep a)"
+ − 1900 shows "OFCLASS('a, at_class)"
+ − 1901 proof
+ − 1902 interpret type_definition Rep Abs "{a. sort_of a \<in> range (\<lambda>x::unit. s)}" by (rule type)
+ − 1903 have sort_of_Rep: "\<And>a. sort_of (Rep a) = s" using Rep by (simp add: image_def)
+ − 1904 fix a b :: 'a and p p1 p2 :: perm
+ − 1905 show "0 \<bullet> a = a"
+ − 1906 unfolding permute_def by (simp add: Rep_inverse)
+ − 1907 show "(p1 + p2) \<bullet> a = p1 \<bullet> p2 \<bullet> a"
+ − 1908 unfolding permute_def by (simp add: Abs_inverse sort_of_Rep)
+ − 1909 show "sort_of (atom a) = sort_of (atom b)"
+ − 1910 unfolding atom_def by (simp add: sort_of_Rep)
+ − 1911 show "atom a = atom b \<longleftrightarrow> a = b"
+ − 1912 unfolding atom_def by (simp add: Rep_inject)
+ − 1913 show "p \<bullet> atom a = atom (p \<bullet> a)"
+ − 1914 unfolding permute_def atom_def by (simp add: Abs_inverse sort_of_Rep)
+ − 1915 qed
+ − 1916 *)
+ − 1917
+ − 1918 lemma at_class:
+ − 1919 fixes s :: atom_sort
+ − 1920 fixes Rep :: "'a \<Rightarrow> atom" and Abs :: "atom \<Rightarrow> 'a"
+ − 1921 assumes type: "type_definition Rep Abs {a. sort_of a = s}"
+ − 1922 assumes atom_def: "\<And>a. atom a = Rep a"
+ − 1923 assumes permute_def: "\<And>p a. p \<bullet> a = Abs (p \<bullet> Rep a)"
+ − 1924 shows "OFCLASS('a, at_class)"
+ − 1925 proof
+ − 1926 interpret type_definition Rep Abs "{a. sort_of a = s}" by (rule type)
+ − 1927 have sort_of_Rep: "\<And>a. sort_of (Rep a) = s" using Rep by (simp add: image_def)
+ − 1928 fix a b :: 'a and p p1 p2 :: perm
+ − 1929 show "0 \<bullet> a = a"
+ − 1930 unfolding permute_def by (simp add: Rep_inverse)
+ − 1931 show "(p1 + p2) \<bullet> a = p1 \<bullet> p2 \<bullet> a"
+ − 1932 unfolding permute_def by (simp add: Abs_inverse sort_of_Rep)
+ − 1933 show "sort_of (atom a) = sort_of (atom b)"
+ − 1934 unfolding atom_def by (simp add: sort_of_Rep)
+ − 1935 show "atom a = atom b \<longleftrightarrow> a = b"
+ − 1936 unfolding atom_def by (simp add: Rep_inject)
+ − 1937 show "p \<bullet> atom a = atom (p \<bullet> a)"
+ − 1938 unfolding permute_def atom_def by (simp add: Abs_inverse sort_of_Rep)
+ − 1939 qed
+ − 1940
+ − 1941 setup {* Sign.add_const_constraint
+ − 1942 (@{const_name "permute"}, SOME @{typ "perm \<Rightarrow> 'a::pt \<Rightarrow> 'a"}) *}
+ − 1943 setup {* Sign.add_const_constraint
+ − 1944 (@{const_name "atom"}, SOME @{typ "'a::at_base \<Rightarrow> atom"}) *}
+ − 1945
+ − 1946
2470
+ − 1947
+ − 1948 section {* The freshness lemma according to Andy Pitts *}
+ − 1949
+ − 1950 lemma freshness_lemma:
+ − 1951 fixes h :: "'a::at \<Rightarrow> 'b::pt"
+ − 1952 assumes a: "\<exists>a. atom a \<sharp> (h, h a)"
+ − 1953 shows "\<exists>x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x"
+ − 1954 proof -
+ − 1955 from a obtain b where a1: "atom b \<sharp> h" and a2: "atom b \<sharp> h b"
+ − 1956 by (auto simp add: fresh_Pair)
+ − 1957 show "\<exists>x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x"
+ − 1958 proof (intro exI allI impI)
+ − 1959 fix a :: 'a
+ − 1960 assume a3: "atom a \<sharp> h"
+ − 1961 show "h a = h b"
+ − 1962 proof (cases "a = b")
+ − 1963 assume "a = b"
+ − 1964 thus "h a = h b" by simp
+ − 1965 next
+ − 1966 assume "a \<noteq> b"
+ − 1967 hence "atom a \<sharp> b" by (simp add: fresh_at_base)
+ − 1968 with a3 have "atom a \<sharp> h b"
+ − 1969 by (rule fresh_fun_app)
+ − 1970 with a2 have d1: "(atom b \<rightleftharpoons> atom a) \<bullet> (h b) = (h b)"
+ − 1971 by (rule swap_fresh_fresh)
+ − 1972 from a1 a3 have d2: "(atom b \<rightleftharpoons> atom a) \<bullet> h = h"
+ − 1973 by (rule swap_fresh_fresh)
+ − 1974 from d1 have "h b = (atom b \<rightleftharpoons> atom a) \<bullet> (h b)" by simp
+ − 1975 also have "\<dots> = ((atom b \<rightleftharpoons> atom a) \<bullet> h) ((atom b \<rightleftharpoons> atom a) \<bullet> b)"
+ − 1976 by (rule permute_fun_app_eq)
+ − 1977 also have "\<dots> = h a"
+ − 1978 using d2 by simp
+ − 1979 finally show "h a = h b" by simp
+ − 1980 qed
+ − 1981 qed
+ − 1982 qed
+ − 1983
+ − 1984 lemma freshness_lemma_unique:
+ − 1985 fixes h :: "'a::at \<Rightarrow> 'b::pt"
+ − 1986 assumes a: "\<exists>a. atom a \<sharp> (h, h a)"
+ − 1987 shows "\<exists>!x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x"
+ − 1988 proof (rule ex_ex1I)
+ − 1989 from a show "\<exists>x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x"
+ − 1990 by (rule freshness_lemma)
+ − 1991 next
+ − 1992 fix x y
+ − 1993 assume x: "\<forall>a. atom a \<sharp> h \<longrightarrow> h a = x"
+ − 1994 assume y: "\<forall>a. atom a \<sharp> h \<longrightarrow> h a = y"
+ − 1995 from a x y show "x = y"
+ − 1996 by (auto simp add: fresh_Pair)
+ − 1997 qed
+ − 1998
+ − 1999 text {* packaging the freshness lemma into a function *}
+ − 2000
+ − 2001 definition
+ − 2002 fresh_fun :: "('a::at \<Rightarrow> 'b::pt) \<Rightarrow> 'b"
+ − 2003 where
+ − 2004 "fresh_fun h = (THE x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x)"
+ − 2005
+ − 2006 lemma fresh_fun_apply:
+ − 2007 fixes h :: "'a::at \<Rightarrow> 'b::pt"
+ − 2008 assumes a: "\<exists>a. atom a \<sharp> (h, h a)"
+ − 2009 assumes b: "atom a \<sharp> h"
+ − 2010 shows "fresh_fun h = h a"
+ − 2011 unfolding fresh_fun_def
+ − 2012 proof (rule the_equality)
+ − 2013 show "\<forall>a'. atom a' \<sharp> h \<longrightarrow> h a' = h a"
+ − 2014 proof (intro strip)
+ − 2015 fix a':: 'a
+ − 2016 assume c: "atom a' \<sharp> h"
+ − 2017 from a have "\<exists>x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x" by (rule freshness_lemma)
+ − 2018 with b c show "h a' = h a" by auto
+ − 2019 qed
+ − 2020 next
+ − 2021 fix fr :: 'b
+ − 2022 assume "\<forall>a. atom a \<sharp> h \<longrightarrow> h a = fr"
+ − 2023 with b show "fr = h a" by auto
+ − 2024 qed
+ − 2025
+ − 2026 lemma fresh_fun_apply':
+ − 2027 fixes h :: "'a::at \<Rightarrow> 'b::pt"
+ − 2028 assumes a: "atom a \<sharp> h" "atom a \<sharp> h a"
+ − 2029 shows "fresh_fun h = h a"
+ − 2030 apply (rule fresh_fun_apply)
+ − 2031 apply (auto simp add: fresh_Pair intro: a)
+ − 2032 done
+ − 2033
+ − 2034 lemma fresh_fun_eqvt:
+ − 2035 fixes h :: "'a::at \<Rightarrow> 'b::pt"
+ − 2036 assumes a: "\<exists>a. atom a \<sharp> (h, h a)"
+ − 2037 shows "p \<bullet> (fresh_fun h) = fresh_fun (p \<bullet> h)"
+ − 2038 using a
+ − 2039 apply (clarsimp simp add: fresh_Pair)
+ − 2040 apply (subst fresh_fun_apply', assumption+)
+ − 2041 apply (drule fresh_permute_iff [where p=p, THEN iffD2])
+ − 2042 apply (drule fresh_permute_iff [where p=p, THEN iffD2])
+ − 2043 apply (simp add: atom_eqvt permute_fun_app_eq [where f=h])
+ − 2044 apply (erule (1) fresh_fun_apply' [symmetric])
+ − 2045 done
+ − 2046
+ − 2047 lemma fresh_fun_supports:
+ − 2048 fixes h :: "'a::at \<Rightarrow> 'b::pt"
+ − 2049 assumes a: "\<exists>a. atom a \<sharp> (h, h a)"
+ − 2050 shows "(supp h) supports (fresh_fun h)"
+ − 2051 apply (simp add: supports_def fresh_def [symmetric])
+ − 2052 apply (simp add: fresh_fun_eqvt [OF a] swap_fresh_fresh)
+ − 2053 done
+ − 2054
+ − 2055 notation fresh_fun (binder "FRESH " 10)
+ − 2056
+ − 2057 lemma FRESH_f_iff:
+ − 2058 fixes P :: "'a::at \<Rightarrow> 'b::pure"
+ − 2059 fixes f :: "'b \<Rightarrow> 'c::pure"
+ − 2060 assumes P: "finite (supp P)"
+ − 2061 shows "(FRESH x. f (P x)) = f (FRESH x. P x)"
+ − 2062 proof -
+ − 2063 obtain a::'a where "atom a \<notin> supp P"
+ − 2064 using P by (rule obtain_at_base)
+ − 2065 hence "atom a \<sharp> P"
+ − 2066 by (simp add: fresh_def)
+ − 2067 show "(FRESH x. f (P x)) = f (FRESH x. P x)"
+ − 2068 apply (subst fresh_fun_apply' [where a=a, OF _ pure_fresh])
+ − 2069 apply (cut_tac `atom a \<sharp> P`)
+ − 2070 apply (simp add: fresh_conv_MOST)
+ − 2071 apply (elim MOST_rev_mp, rule MOST_I, clarify)
2479
+ − 2072 apply (simp add: permute_fun_def permute_pure fun_eq_iff)
2470
+ − 2073 apply (subst fresh_fun_apply' [where a=a, OF `atom a \<sharp> P` pure_fresh])
+ − 2074 apply (rule refl)
+ − 2075 done
+ − 2076 qed
+ − 2077
+ − 2078 lemma FRESH_binop_iff:
+ − 2079 fixes P :: "'a::at \<Rightarrow> 'b::pure"
+ − 2080 fixes Q :: "'a::at \<Rightarrow> 'c::pure"
+ − 2081 fixes binop :: "'b \<Rightarrow> 'c \<Rightarrow> 'd::pure"
+ − 2082 assumes P: "finite (supp P)"
+ − 2083 and Q: "finite (supp Q)"
+ − 2084 shows "(FRESH x. binop (P x) (Q x)) = binop (FRESH x. P x) (FRESH x. Q x)"
+ − 2085 proof -
+ − 2086 from assms have "finite (supp P \<union> supp Q)" by simp
+ − 2087 then obtain a::'a where "atom a \<notin> (supp P \<union> supp Q)"
+ − 2088 by (rule obtain_at_base)
+ − 2089 hence "atom a \<sharp> P" and "atom a \<sharp> Q"
+ − 2090 by (simp_all add: fresh_def)
+ − 2091 show ?thesis
+ − 2092 apply (subst fresh_fun_apply' [where a=a, OF _ pure_fresh])
+ − 2093 apply (cut_tac `atom a \<sharp> P` `atom a \<sharp> Q`)
+ − 2094 apply (simp add: fresh_conv_MOST)
+ − 2095 apply (elim MOST_rev_mp, rule MOST_I, clarify)
2479
+ − 2096 apply (simp add: permute_fun_def permute_pure fun_eq_iff)
2470
+ − 2097 apply (subst fresh_fun_apply' [where a=a, OF `atom a \<sharp> P` pure_fresh])
+ − 2098 apply (subst fresh_fun_apply' [where a=a, OF `atom a \<sharp> Q` pure_fresh])
+ − 2099 apply (rule refl)
+ − 2100 done
+ − 2101 qed
+ − 2102
+ − 2103 lemma FRESH_conj_iff:
+ − 2104 fixes P Q :: "'a::at \<Rightarrow> bool"
+ − 2105 assumes P: "finite (supp P)" and Q: "finite (supp Q)"
+ − 2106 shows "(FRESH x. P x \<and> Q x) \<longleftrightarrow> (FRESH x. P x) \<and> (FRESH x. Q x)"
+ − 2107 using P Q by (rule FRESH_binop_iff)
+ − 2108
+ − 2109 lemma FRESH_disj_iff:
+ − 2110 fixes P Q :: "'a::at \<Rightarrow> bool"
+ − 2111 assumes P: "finite (supp P)" and Q: "finite (supp Q)"
+ − 2112 shows "(FRESH x. P x \<or> Q x) \<longleftrightarrow> (FRESH x. P x) \<or> (FRESH x. Q x)"
+ − 2113 using P Q by (rule FRESH_binop_iff)
+ − 2114
+ − 2115
2467
+ − 2116 section {* Library functions for the nominal infrastructure *}
+ − 2117
1833
2050b5723c04
added a library for basic nominal functions; separated nominal_eqvt file
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 2118 use "nominal_library.ML"
2050b5723c04
added a library for basic nominal functions; separated nominal_eqvt file
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 2119
2466
+ − 2120
2467
+ − 2121 section {* Automation for creating concrete atom types *}
+ − 2122
+ − 2123 text {* at the moment only single-sort concrete atoms are supported *}
+ − 2124
+ − 2125 use "nominal_atoms.ML"
+ − 2126
+ − 2127
2466
+ − 2128
1062
+ − 2129 end